Planck 2015 results. XIII. Cosmological parameters · 2016-06-20 · Astronomy & Astrophysics manuscript no. planck˙parameters˙2015 c ESO 2016 June 20, 2016 Planck 2015 results

Astronomy & Astrophysics manuscript no. planck˙parameters˙2015 c© ESO 2016June 20, 2016

Planck 2015 results. XIII. Cosmological parametersPlanck Collaboration: P. A. R. Ade105, N. Aghanim71, M. Arnaud87, M. Ashdown83,7, J. Aumont71, C. Baccigalupi103, A. J. Banday117,12,R. B. Barreiro78, J. G. Bartlett1,80, N. Bartolo38,79, E. Battaner120,121, R. Battye81, K. Benabed72,116, A. Benoıt69, A. Benoit-Levy29,72,116,

J.-P. Bernard117,12, M. Bersanelli41,58, P. Bielewicz97,12,103, J. J. Bock80,14, A. Bonaldi81, L. Bonavera78, J. R. Bond11, J. Borrill17,109,F. R. Bouchet72,107, F. Boulanger71, M. Bucher1, C. Burigana57,39,59, R. C. Butler57, E. Calabrese112, J.-F. Cardoso88,1,72, A. Catalano89,86,

A. Challinor75,83,15, A. Chamballu87,19,71, R.-R. Chary68, H. C. Chiang33,8, J. Chluba28,83, P. R. Christensen98,44, S. Church111, D. L. Clements67,S. Colombi72,116, L. P. L. Colombo27,80, C. Combet89, A. Coulais86, B. P. Crill80,14, A. Curto78,7,83, F. Cuttaia57, L. Danese103, R. D. Davies81,

R. J. Davis81, P. de Bernardis40, A. de Rosa57, G. de Zotti54,103, J. Delabrouille1, F.-X. Desert64, E. Di Valentino72,107, C. Dickinson81,J. M. Diego78, K. Dolag119,94, H. Dole71,70, S. Donzelli58, O. Dore80,14, M. Douspis71, A. Ducout72,67, J. Dunkley112, X. Dupac47,

G. Efstathiou75,83∗, F. Elsner29,72,116, T. A. Enßlin94, H. K. Eriksen76, M. Farhang11,102, J. Fergusson15, F. Finelli57,59, O. Forni117,12, M. Frailis56,A. A. Fraisse33, E. Franceschi57, A. Frejsel98, S. Galeotta56, S. Galli82, K. Ganga1, C. Gauthier1,93, M. Gerbino114,100,40, T. Ghosh71, M. Giard117,12,Y. Giraud-Heraud1, E. Giusarma40, E. Gjerløw76, J. Gonzalez-Nuevo23,78, K. M. Gorski80,123, S. Gratton83,75, A. Gregorio42,56,63, A. Gruppuso57,

J. E. Gudmundsson114,100,33, J. Hamann115,113, F. K. Hansen76, D. Hanson95,80,11, D. L. Harrison75,83, G. Helou14, S. Henrot-Versille85,C. Hernandez-Monteagudo16,94, D. Herranz78, S. R. Hildebrandt80,14, E. Hivon72,116, M. Hobson7, W. A. Holmes80, A. Hornstrup20, W. Hovest94,

Z. Huang11, K. M. Huffenberger31, G. Hurier71, A. H. Jaffe67, T. R. Jaffe117,12, W. C. Jones33, M. Juvela32, E. Keihanen32, R. Keskitalo17,T. S. Kisner91, R. Kneissl46,9, J. Knoche94, L. Knox35, M. Kunz21,71,3, H. Kurki-Suonio32,52, G. Lagache5,71, A. Lahteenmaki2,52, J.-M. Lamarre86,

A. Lasenby7,83, M. Lattanzi39,60, C. R. Lawrence80, J. P. Leahy81, R. Leonardi10, J. Lesgourgues73,115, F. Levrier86, A. Lewis30, M. Liguori38,79,P. B. Lilje76, M. Linden-Vørnle20, M. Lopez-Caniego47,78, P. M. Lubin36, J. F. Macıas-Perez89, G. Maggio56, D. Maino41,58, N. Mandolesi57,39,

A. Mangilli71,85, A. Marchini61, M. Maris56, P. G. Martin11, M. Martinelli122, E. Martınez-Gonzalez78, S. Masi40, S. Matarrese38,79,49,P. McGehee68, P. R. Meinhold36, A. Melchiorri40,61, J.-B. Melin19, L. Mendes47, A. Mennella41,58, M. Migliaccio75,83, M. Millea35, S. Mitra66,80,

M.-A. Miville-Deschenes71,11, A. Moneti72, L. Montier117,12, G. Morgante57, D. Mortlock67, A. Moss106, D. Munshi105, J. A. Murphy96,P. Naselsky99,45, F. Nati33, P. Natoli39,4,60, C. B. Netterfield24, H. U. Nørgaard-Nielsen20, F. Noviello81, D. Novikov92, I. Novikov98,92,

C. A. Oxborrow20, F. Paci103, L. Pagano40,61, F. Pajot71, R. Paladini68, D. Paoletti57,59, B. Partridge51, F. Pasian56, G. Patanchon1, T. J. Pearson14,68,O. Perdereau85, L. Perotto89, F. Perrotta103, V. Pettorino50, F. Piacentini40, M. Piat1, E. Pierpaoli27, D. Pietrobon80, S. Plaszczynski85,

E. Pointecouteau117,12, G. Polenta4,55, L. Popa74, G. W. Pratt87, G. Prezeau14,80, S. Prunet72,116, J.-L. Puget71, J. P. Rachen25,94, W. T. Reach118,R. Rebolo77,18,22, M. Reinecke94, M. Remazeilles81,71,1, C. Renault89, A. Renzi43,62, I. Ristorcelli117,12, G. Rocha80,14, C. Rosset1, M. Rossetti41,58,

G. Roudier1,86,80, B. Rouille d’Orfeuil85, M. Rowan-Robinson67, J. A. Rubino-Martın77,22, B. Rusholme68, N. Said40, V. Salvatelli40,6, L. Salvati40,M. Sandri57, D. Santos89, M. Savelainen32,52, G. Savini101, D. Scott26, M. D. Seiffert80,14, P. Serra71, E. P. S. Shellard15, L. D. Spencer105,

M. Spinelli85, V. Stolyarov7,110,84, R. Stompor1, R. Sudiwala105, R. Sunyaev94,108, D. Sutton75,83, A.-S. Suur-Uski32,52, J.-F. Sygnet72,J. A. Tauber48, L. Terenzi104,57, L. Toffolatti23,78,57, M. Tomasi41,58, M. Tristram85, T. Trombetti57,39, M. Tucci21, J. Tuovinen13, M. Turler65,G. Umana53, L. Valenziano57, J. Valiviita32,52, F. Van Tent90, P. Vielva78, F. Villa57, L. A. Wade80, B. D. Wandelt72,116,37, I. K. Wehus80,76,

M. White34, S. D. M. White94, A. Wilkinson81, D. Yvon19, A. Zacchei56, and A. Zonca36

(Affiliations can be found after the references)

June 20, 2016ABSTRACT

This paper presents cosmological results based on full-mission Planck observations of temperature and polarization anisotropies of the cosmicmicrowave background (CMB) radiation. Our results are in very good agreement with the 2013 analysis of the Planck nominal-mission tempera-ture data, but with increased precision. The temperature and polarization power spectra are consistent with the standard spatially-flat 6-parameterΛCDM cosmology with a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper). From the Planck tempera-ture data combined with Planck lensing, for this cosmology we find a Hubble constant, H0 = (67.8±0.9) km s−1Mpc−1, a matter density parameterΩm = 0.308±0.012, and a tilted scalar spectral index with ns = 0.968±0.006, consistent with the 2013 analysis. Note that in this abstract we quote68 % confidence limits on measured parameters and 95 % upper limits on other parameters. We present the first results of polarization measure-ments with the Low Frequency Instrument at large angular scales. Combined with the Planck temperature and lensing data, these measurementsgive a reionization optical depth of τ = 0.066 ± 0.016, corresponding to a reionization redshift of zre = 8.8+1.7

−1.4. These results are consistent withthose from WMAP polarization measurements cleaned for dust emission using 353-GHz polarization maps from the High Frequency Instrument.We find no evidence for any departure from base ΛCDM in the neutrino sector of the theory; for example, combining Planck observations withother astrophysical data we find Neff = 3.15±0.23 for the effective number of relativistic degrees of freedom, consistent with the value Neff = 3.046of the Standard Model of particle physics. The sum of neutrino masses is constrained to

∑mν < 0.23 eV. The spatial curvature of our Universe is

found to be very close to zero, with |ΩK | < 0.005. Adding a tensor component as a single-parameter extension to base ΛCDM we find an upperlimit on the tensor-to-scalar ratio of r0.002 < 0.11, consistent with the Planck 2013 results and consistent with the B-mode polarization constraintsfrom a joint analysis of BICEP2, Keck Array, and Planck (BKP) data. Adding the BKP B-mode data to our analysis leads to a tighter constraint ofr0.002 < 0.09 and disfavours inflationary models with a V(φ) ∝ φ2 potential. The addition of Planck polarization data leads to strong constraints ondeviations from a purely adiabatic spectrum of fluctuations. We find no evidence for any contribution from isocurvature perturbations or from cos-mic defects. Combining Planck data with other astrophysical data, including Type Ia supernovae, the equation of state of dark energy is constrainedto w = −1.006 ± 0.045, consistent with the expected value for a cosmological constant. The standard big bang nucleosynthesis predictions for thehelium and deuterium abundances for the best-fit Planck base ΛCDM cosmology are in excellent agreement with observations. We also analyseconstraints on annihilating dark matter and on possible deviations from the standard recombination history. In neither case do we find no evidencefor new physics. The Planck results for base ΛCDM are in good agreement with baryon acoustic oscillation data and with the JLA sample of TypeIa supernovae. However, as in the 2013 analysis, the amplitude of the fluctuation spectrum is found to be higher than inferred from some analysesof rich cluster counts and weak gravitational lensing. We show that these tensions cannot easily be resolved with simple modifications of the baseΛCDM cosmology. Apart from these tensions, the base ΛCDM cosmology provides an excellent description of the Planck CMB observations andmany other astrophysical data sets.

Key words. Cosmology: observations – Cosmology: theory – cosmic microwave background – cosmological parameters 1

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1. Introduction

The cosmic microwave background (CMB) radiation offers anextremely powerful way of testing the origin of fluctuations andof constraining the matter content, geometry, and late-time evo-lution of the Universe. Following the discovery of anisotropiesin the CMB by the COBE satellite (Smoot et al. 1992), ground-based, sub-orbital experiments and notably the WMAP satellite(Bennett et al. 2003, 2013) have mapped the CMB anisotropieswith increasingly high precision, providing a wealth of new in-formation on cosmology.

Planck1 is the third-generation space mission, follow-ing COBE and WMAP, dedicated to measurements of theCMB anisotropies. The first cosmological results from Planckwere reported in a series of papers (for an overview seePlanck Collaboration I 2014, and references therein) togetherwith a public release of the first 15.5 months of temperaturedata (which we will refer to as the nominal mission data).Constraints on cosmological parameters from Planck were re-ported in Planck Collaboration XVI (2014).2 The Planck 2013analysis showed that the temperature power spectrum fromPlanck was remarkably consistent with a spatially flat ΛCDMcosmology specified by six parameters, which we will refer toas the base ΛCDM model. However, the cosmological param-eters of this model were found to be in tension, typically atthe 2–3σ level, with some other astronomical measurements,most notably direct estimates of the Hubble constant (Riess et al.2011), the matter density determined from distant supernovae(Conley et al. 2011; Rest et al. 2014), and estimates of the am-plitude of the fluctuation spectrum from weak gravitationallensing (Heymans et al. 2013; Mandelbaum et al. 2013) and theabundance of rich clusters of galaxies (Planck Collaboration XX2014; Benson et al. 2013; Hasselfield et al. 2013). As reported inthe revised version of PCP13, and discussed further in Sect. 5,some of these tensions have been resolved with the acquisition ofmore astrophysical data, while other new tensions have emerged.

The primary goal of this paper is to present the results fromthe full Planck mission, including a first analysis of the Planckpolarization data. In addition, this paper introduces some refine-ments in data analysis and addresses the effects of small in-strumental systematics discovered (or better understood) sincePCP13 appeared.

The Planck 2013 data were not entirely free of systematiceffects. The Planck instruments and analysis chains are com-plex and our understanding of systematics has improved sincePCP13. The most important of these was the incomplete re-moval of line-like features in the power spectrum of the time-ordered data, caused by interference of the 4-K cooler electron-ics with the bolometer readout electronics. This resulted in cor-related systematics across detectors, leading to a small “dip” inthe power spectra at multipoles ` ≈ 1800 at 217 GHz, which is

∗Corresponding author: G. Efstathiou, [email protected] (http://www.esa.int/Planck) is a project of the

European Space Agency (ESA) with instruments provided by two sci-entific consortia funded by ESA member states and led by PrincipalInvestigators from France and Italy, telescope reflectors providedthrough a collaboration between ESA and a scientific consortium ledand funded by Denmark, and additional contributions from NASA(USA).

2This paper refers extensively to the earlier 2013 Planck cosmo-logical parameters paper and CMB power spectra and likelihood paper(Planck Collaboration XVI 2014; Planck Collaboration XV 2014). Tosimplify the presentation, these papers will henceforth be referred to asPCP13 and PPL13, respectively.

most noticeable in the first sky survey. Various tests were pre-sented in PCP13 that suggested that this systematic caused onlysmall shifts to cosmological parameters. Further analyses, basedon the full mission data from the HFI (29 months, 4.8 sky sur-veys) are consistent with this conclusion (see Sect. 3). In addi-tion, we discovered a minor error in the beam transfer functionsapplied to the 2013 217-GHz spectra, which had negligible im-pact on the scientific results. Another feature of the Planck data,not fully understood at the time of the 2013 data release, was a2.6 % calibration offset (in power) between Planck and WMAP(reported in PCP13, see also Planck Collaboration XXXI 2014).As discussed in Appendix A of PCP13, the 2013 Planck andWMAP power spectra agree to high precision if this multiplica-tive factor is taken into account and it has no significant im-pact on cosmological parameters apart from a rescaling of theamplitude of the primordial fluctuation spectrum. The reasonsfor the 2013 calibration offsets are now largely understood andin the 2015 release the calibrations of both Planck instrumentsand WMAP are consistent to within about 0.3 % in power (seePlanck Collaboration I 2016, for further details). In addition, thePlanck beams have been characterized more accurately in the2015 data release and there have been minor modifications tothe low-level data processing.

The layout of this paper is as follows. Section 2 summarizesa number of small changes to the parameter estimation method-ology since PCP13. The full mission temperature and polariza-tion power spectra are presented in Sect. 3. The first subsection(Sect. 3.1) discusses the changes in the cosmological parametersof the base ΛCDM cosmology compared to those presented in2013. Section 3.2 presents an assessment of the impact of fore-ground cleaning (using the 545-GHz maps) on the cosmologicalparameters of the base ΛCDM model. The power spectra andassociated likelihoods are presented in Sect. 3.3. This subsec-tion also discusses the internal consistency of the Planck TT ,T E, and EE spectra. The agreement of T E and EE with the TTspectra provides an important additional test of the accuracy ofour foreground corrections to the TT spectra at high multipoles.

PCP13 used the WMAP polarization likelihood at low mul-tipoles to constrain the reionization optical depth parameter τ.The 2015 analysis replaces the WMAP likelihood with polar-ization data from the Planck Low Frequency Instrument (LFI,Planck Collaboration II 2016). The impact of this change on τ isdiscussed in Sect. 3.4, which also presents an alternative (andcompetitive) constraint on τ based on combining the PlanckTT spectrum with the power spectrum of the lensing poten-tial measured by Planck. We also compare the LFI polarizationconstraints with the WMAP polarization data cleaned with thePlanck HFI 353-GHz maps.

Section 4 compares the Planck power spectra with the powerspectra from high-resolution ground-based CMB data from theAtacama Cosmology Telescope (ACT, Das et al. 2014) and theSouth Pole Telescope (SPT, George et al. 2015). This sectionapplies a Gibbs sampling technique to sample over foregroundand other “nuisance” parameters to recover the underlyingCMB power spectrum at high multipoles (Dunkley et al. 2013;Calabrese et al. 2013). Unlike PCP13, in which we combined thelikelihoods of the high-resolution experiments with the Plancktemperature likelihood, in this paper we use the high-resolutionexperiments mainly to check the consistency of the “dampingtail” in the Planck power spectrum at multipoles >∼ 2000.

Section 5 introduces additional data, includingthe Planck lensing likelihood (described in detail inPlanck Collaboration XV 2016) and other astrophysical datasets. As in PCP13, we are highly selective in the astrophysical

[email protected]

http://www.esa.int/Planck

Planck Collaboration: Cosmological parameters

data sets that we combine with Planck. As mentioned above, themain purpose of this paper is to describe what the Planck datahave to say about cosmology. It is not our purpose to present anexhaustive discussion of what happens when the Planck data arecombined with a wide range of astrophysical data. This can bedone by others, using the publicly released Planck likelihood.Nevertheless, some cosmological parameter combinations arehighly degenerate using CMB power spectrum measurementsalone, the most severe being the “geometrical degeneracy” thatopens up when spatial curvature is allowed to vary. Baryonacoustic oscillation (BAO) measurements are a particularlyimportant astrophysical data set. Since BAO surveys involvea simple geometrical measurement, these data are less proneto systematic errors than most other astrophysical data. As inPCP13, BAO measurements are used as a primary astrophys-ical data set in combination with Planck to break parameterdegeneracies. It is worth mentioning explicitly our approach tointerpreting tensions between Planck and other astrophysicaldata sets. Tensions may be indicators of new physics beyondthat assumed in the base ΛCDM model. However, they may alsobe caused by systematic errors in the data. Our primary goalis to report whether the Planck data support any evidence fornew physics. If evidence for new physics is driven primarily byastrophysical data, but not by Planck, then the emphasis mustnecessarily shift to establishing whether the astrophysical dataare free of systematics. This type of assessment is beyond thescope of this paper, but sets a course for future research.

Extensions to the base ΛCDM cosmology are discussed inSect. 6, which explores a large grid of possibilities. In additionto these models, we also explore constraints on big bang nu-cleosynthesis, dark matter annihilation, cosmic defects, and de-partures from the standard recombination history. As in PCP13,we find no convincing evidence for a departure from the baseΛCDM model. As far as we can tell, a simple inflationary modelwith a slightly tilted, purely adiabatic, scalar fluctuation spec-trum fits the Planck data and most other precision astrophys-ical data. There are some “anomalies” in this picture, includ-ing the poor fit to the CMB temperature fluctuation spectrumat low multipoles, as reported by WMAP (Bennett et al. 2003)and in PCP13, suggestions of departures from statistical isotropyat low multipoles (as reviewed in Planck Collaboration XXIII2014; Planck Collaboration XVI 2016), and hints of a discrep-ancy with the amplitude of the matter fluctuation spectrum atlow redshifts (see Sect. 5.5). However, none of these anomaliesare of decisive statistical significance at this stage.

One of the most interesting developments since the ap-pearance of PCP13 was the detection by the BICEP2 teamof a B-mode polarization anisotropy (BICEP2 Collaboration2014), apparently in conflict with the 95 % upper limiton the tensor-to-scalar ratio, r0.002 < 0.11,3 reported inPCP13. Clearly, the detection of B-mode signal from pri-mordial gravitational waves would have profound conse-quences for cosmology and inflationary theory. However, anumber of studies, in particular an analysis of Planck 353-GHz polarization data, suggested that polarized dust emis-sion might contribute a significant part of the BICEP2 sig-nal (Planck Collaboration Int. XXX 2016; Mortonson & Seljak

3The subscript on r refers to the pivot scale in Mpc−1 used to de-fine the tensor-to-scalar ratio. For Planck we usually quote r0.002, sincea pivot scale of 0.002 Mpc−1 is close to the scale at which there is somesensitivity to tensor modes in the large-angle temperature power spec-trum. For a scalar spectrum with no running and a scalar spectral indexof ns = 0.965, r0.05 ≈ 1.12r0.002 for small r. For r ≈ 0.1, assuming theinflationary consistency relation, we have instead r0.05 ≈ 1.08r0.002.

2014; Flauger et al. 2014). The situation is now clearer fol-lowing the joint analysis of BICEP2, Keck Array, andPlanck data (BICEP2/Keck Array and Planck Collaborations2015, hereafter BKP); this increases the signal-to-noise ratio onpolarized dust emission primarily by directly cross-correlatingthe BICEP2 and Keck Array data at 150 GHz with the Planckpolarization data at 353 GHz. The results of BKP give a 95 %upper limit on the tensor-to-scalar ratio of r0.05 < 0.12, withno statistically significant evidence for a primordial gravitationalwave signal. Section 6.2 presents a brief discussion of this resultand how it fits in with the indirect constraints on r derived fromthe Planck 2015 data.

Our conclusions are summarized in Sect. 7.

2. Model, parameters, and methodology

The notation, definitions and methodology used in this paperlargely follow those described in PCP13, and so will not be re-peated here. For completeness, we list some derived parametersof interest in Sect. 2.2. We have made a small number of modi-fications to the methodology, as described in Sect. 2.1. We havealso made some minor changes to the model of unresolved fore-grounds and nuisance parameters used in the high-` likelihood.These are described in detail in Planck Collaboration XI (2016),but to make this paper more self-contained, these changes aresummarized in Sect. 2.3.

2.1. Theoretical model

We adopt the same general methodology as described in PCP13,with small modifications. Our main results are now based on thelensed CMB power spectra computed with the updated January2015 version of the camb4 Boltzmann code (Lewis et al. 2000),and parameter constraints are based on the January 2015 versionof CosmoMC (Lewis & Bridle 2002; Lewis 2013). Changes inour physical modelling are as follows.

• For each model in which the fraction of baryonic mass inhelium YP is not varied independently of other parameters,it is now set from the big bang nucleosynthesis (BBN) pre-diction by interpolation from a recent fitting formula basedon results from the PArthENoPE BBN code (Pisanti et al.2008). We now use a fixed fiducial neutron decay constantof τn = 880.3 s, and also account for the small difference be-tween the mass-fraction ratio YP and the nucleon-based frac-tion YBBN

P . These modifications result in changes of about1 % to the inferred value of YP compared to PCP13, givingbest-fit values YP ≈ 0.2453 (YBBN

P ≈ 0.2467) in ΛCDM.See Sect. 6.5 for a detailed discussion of the impact of un-certainties arising from variations of τn and nuclear reac-tion rates; however, these uncertainties have minimal impacton our main results. Section 6.5 also corrects a small errorarising from how the difference between Neff = 3.046 andNeff = 3 was handled in the BBN fitting formula.

• We have corrected a missing source term in the dark energymodelling for w , −1. The correction of this error has verylittle impact on our science results, since it is only importantfor values of w far from −1.

• To model the small-scale matter power spectrum, we use thehalofit approach (Smith et al. 2003), with the updates ofTakahashi et al. (2012), as in PCP13, but with revised fitting

4http://camb.info

3

http://camb.info


parameters for massive neutrino models.5 We also now in-clude the halofit corrections when calculating the lensedCMB power spectra.

As in PCP13 we adopt a Bayesian framework for testingtheoretical models. Tests using the “profile likelihood” method,described in Planck Collaboration Int. XVI (2014), show excel-lent agreement for the mean values of the cosmological pa-rameters and their errors, for both the base ΛCDM model andits Neff extension. Tests have also been carried out using theclass Boltzmann code (Lesgourgues 2011) and the MontePythonMCMC code (Audren et al. 2013) in place of camb andCosmoMC, respectively. Again, for flat models we find excellentagreement with the baseline choices used in this paper.

2.2. Derived parameters

Our base parameters are defined as in PCP13, and we also calcu-late the same derived parameters. In addition we now compute:

• the helium nucleon fraction defined by YBBNP ≡ 4nHe/nb;

• where standard BBN is assumed, the mid-value deuteriumratio predicted by BBN, yDP ≡ 105nD/nH, using a fit fromthe PArthENoPE BBN code (Pisanti et al. 2008);

• the comoving wavenumber of the perturbation mode thatentered the Hubble radius at matter-radiation equality zeq,where this redshift is calculated approximating all neutrinosas relativistic at that time, i.e., keq ≡ a(zeq)H(zeq);

• the comoving angular diameter distance to last scattering,DA(z∗);

• the angular scale of the sound horizon at matter-radiationequality, θs,eq ≡ rs(zeq)/DA(z∗), where rs is the sound hori-zon and z∗ is the redshift of last scattering;

• the amplitude of the CMB power spectrum D` ≡ `(` +1)C`/2π in µK2, for ` = 40, 220, 810, 1520, and 2000;

• the primordial spectral index of the curvature perturbationsat wavenumber k = 0.002 Mpc−1, ns,0.002 (as in PCP13, ourdefault pivot scale is k = 0.05 Mpc−1, so that ns ≡ ns,0.05);

• parameter combinations close to those probed by galaxy andCMB lensing (and other external data), specifically σ8Ω0.5

mand σ8Ω0.25

m ;• various quantities reported by BAO and redshift-space dis-

tortion measurements, as described in Sects. 5.2 and 5.5.1.

2.3. Changes to the foreground model

Unresolved foregrounds contribute to the temperature powerspectrum and must be modelled to extract accurate cosmolog-ical parameters. PPL13 and PCP13 used a parametric approachto modelling foregrounds, similar to the approach adopted in theanalysis of the SPT and ACT experiments (Reichardt et al. 2012;Dunkley et al. 2013). The unresolved foregrounds are describedby a set of power spectrum templates together with nuisance pa-rameters, which are sampled via MCMC along with the cosmo-logical parameters.6 The components of the extragalactic fore-ground model consist of:

5Results for neutrino models with galaxy and CMB lensing aloneuse the camb Jan 2015 version of halofit to avoid problems at largeΩm; other results use the previous (April 2014) halofit version.

6Our treatment of Galactic dust emission also differs from that usedin PPL13 and PCP13. Here we describe changes to the extragalacticmodel and our treatment of errors in the Planck absolute calibration,deferring a discussion of Galactic dust modelling in temperature andpolarization to Sect. 3.

• the shot noise from Poisson fluctuations in the number den-sity of point sources;

• the power due to clustering of point sources (loosely referredto as the CIB component);

• a thermal Sunyaev-Zeldovich (tSZ) component;• a kinetic Sunyaev-Zeldovich (kSZ) component;• the cross-correlation between tSZ and CIB.

In addition, the likelihood includes a number of other nui-sance parameters, such as relative calibrations between frequen-cies, and beam eigenmode amplitudes. We use the same tem-plates for the tSZ, kSZ, and tSZ/CIB cross-correlation as in the2013 papers. However, we have made a number of changes to theCIB modelling and the priors adopted for the SZ effects, whichwe now describe in detail.

2.3.1. CIB

In the 2013 papers, the CIB anisotropies were modelled as apower law:

Dν1×ν2`

= ACIBν1×ν2

(`

3000

)γCIB

. (1)

Planck data alone provide a constraint on ACIB217×217 and very weak

constraints on the CIB amplitudes at lower frequencies. PCP13reported typical values of ACIB

217×217 = (29 ± 6) µK2 and γCIB =0.40 ± 0.15, fitted over the range 500 ≤ ` ≤ 2500. The additionof the ACT and SPT data (“highL”) led to solutions with steepervalues of γCIB, closer to 0.8, suggesting that the CIB componentwas not well fit by a power law.

Planck results on the CIB, using H i as a tracer of Galacticdust, are discussed in detail in Planck Collaboration XXX(2014). In that paper, a model with 1-halo and 2-halo con-tributions was developed that provides an accurate descriptionof the Planckand IRAS CIB spectra from 217 GHz through to3000 GHz. At high multipoles, ` >∼ 3000, the halo-model spectraare reasonably well approximated by power laws, with a slopeγCIB ≈ 0.8 (though see Sect. 4). At multipoles in the range500 <∼ ` <∼ 2000, corresponding to the transition from the 2-haloterm dominating the clustering power to the 1-halo term domi-nating, the Planck Collaboration XXX (2014) templates have ashallower slope, consistent with the results of PCP13. The am-plitudes of these templates at ` = 3000 are

ACIB217×217 = 63.6 µK2, ACIB

143×217 = 19.1, µK2,

ACIB143×143 = 5.9 µK2, ACIB

100×100 = 1.4 µK2. (2)

Note that in PCP13, the CIB amplitude of the 143×217 spectrumwas characterized by a correlation coefficient

ACIB143×217 = rCIB

143×217

√ACIB

217×217ACIB143×143. (3)

The combined Planck+highL solutions in PCP13 always give ahigh correlation coefficient with a 95 % lower limit of rCIB

143×217>∼

0.85, consistent with the model of Eq. (2), which has rCIB143×217 ≈

1. In the 2015 analysis, we use the Planck Collaboration XXX(2014) templates, fixing the relative amplitudes at 100 × 100,143 × 143, and 143 × 217 to the amplitude of the 217 × 217spectrum. Thus, the CIB model used in this paper is specified byonly one amplitude, ACIB

217×217, which is assigned a uniform priorin the range 0–200 µK2.

4


In PCP13 we solved for the CIB amplitudes at the CMBeffective frequencies of 217 and 143 GHz, and so we includedcolour corrections in the amplitudes ACIB

217×217 and ACIB143×143 (there

was no CIB component in the 100 × 100 spectrum). In the 2015Planck analysis, we do not include a colour term since we defineACIB

217×217 to be the actual CIB amplitude measured in the Planck217-GHz band. This is higher by a factor of about 1.33 com-pared to the amplitude at the CMB effective frequency of thePlanck 217-GHz band. This should be borne in mind by readerscomparing 2015 and 2013 CIB amplitudes measured by Planck.

2.3.2. Thermal and kinetic SZ amplitudes

In the 2013 papers we assumed template shapes for the thermal(tSZ) and kinetic (kSZ) spectra characterized by two amplitudes,AtSZ and AkSZ, defined in equations (26) and (27) of PCP13.These amplitudes were assigned uniform priors in the range 0–10 (µK)2 . We used the Trac et al. (2011) kSZ template spec-trum and the ε = 0.5 tSZ template from Efstathiou & Migliaccio(2012). We adopt the same templates for the 2015 Planckanalysis, since, for example, the tSZ template is actually agood match to the results from the recent numerical simula-tions of McCarthy et al. (2014). In addition, we previously in-cluded a template from Addison et al. (2012) to model the cross-correlation between the CIB and tSZ emission from clusters ofgalaxies. The amplitude of this template was characterized bya dimensionless correlation coefficient, ξtSZ×CIB, which was as-signed a uniform prior in the range 0–1. The three parametersAtSZ, AkSZ, and ξtSZ×CIB, are not well constrained by Planckalone. Even when combined with ACT and SPT, the three pa-rameters are highly correlated with each other. Marginalizingover ξtSZ×CIB, Reichardt et al. (2012) find that SPT spectra con-strain the linear combination

AkSZ + 1.55 AtSZ = (9.2 ± 1.3) µK2. (4)

The slight differences in the coefficients compared to the formulagiven in Reichardt et al. (2012) come from the different effec-tive frequencies used to define the Planck amplitudes AkSZ andAtSZ. An investigation of the 2013 Planck+highL solutions showa similar degeneracy direction, which is almost independent ofcosmology, even for extensions to the base ΛCDM model:

ASZ = AkSZ + 1.6 AtSZ = (9.4 ± 1.4) µK2 (5)

for Planck+WP+highL, which is very close to the degener-acy direction (Eq. 4) measured by SPT. In the 2015 Planckanalysis, we impose a conservative Gaussian prior for ASZ, asdefined in Eq. (5), with a mean of 9.5 µK2 and a dispersion3µK2 (i.e., somewhat broader than the dispersion measured byReichardt et al. 2012). The purpose of imposing this prior on ASZ

is to prevent the parameters AkSZ and AtSZ from wandering intounphysical regions of parameter space when using Planck dataalone. We retain the uniform prior of [0,1] for ξtSZ×CIB. As thispaper was being written, results from the complete 2540 deg2

SPT-SZ survey area appeared (George et al. 2015). These areconsistent with Eq. (5) and in addition constrain the correla-tion parameter to low values, ξtSZ×CIB = 0.113+0.057

−0.054. The looserpriors on these parameters adopted in this paper are, however,sufficient to eliminate any significant sensitivity of cosmologi-cal parameters derived from Planck to the modelling of the SZcomponents.

2.3.3. Absolute Planck calibration

In PCP13, we treated the calibrations of the 100 and 217-GHzchannels relative to 143 GHz as nuisance parameters. This wasan approximate way of dealing with small differences in rela-tive calibrations between different detectors at high multipoles,caused by bolometer time-transfer function corrections and in-termediate and far sidelobes of the Planck beams. In otherwords, we approximated these effects as a purely multiplicativecorrection to the power spectra over the multipole range ` = 50–2500. The absolute calibration of the 2013 Planck power spectrawas therefore fixed, by construction, to the absolute calibrationof the 143-5 bolometer. Any error in the absolute calibration ofthis reference bolometer was not propagated into errors on cos-mological parameters. For the 2015 Planck likelihoods we usean identical relative calibration scheme between 100, 143, and217 GHz, but we now include an absolute calibration parame-ter yp, at the map level, for the 143-GHz reference frequency.We adopt a Gaussian prior on yp centred on unity with a (con-servative) dispersion of 0.25 %. This overall calibration uncer-tainty is then propagated through to cosmological parameterssuch as As and σ8. A discussion of the consistency of the abso-lute calibrations across the nine Planck frequency bands is givenin Planck Collaboration I (2016).

3. Constraints on the parameters of the baseΛCDM cosmology from Planck

3.1. Changes in the base ΛCDM parameters compared tothe 2013 data release

The principal conclusion of PCP13 was the excellent agreementof the base ΛCDM model with the temperature power spectrameasured by Planck. In this subsection, we compare the param-eters of the base ΛCDM model reported in PCP13 with thosemeasured from the full-mission 2015 data. Here we restrict thecomparison to the high multipole temperature (TT ) likelihood(plus low-` polarization), postponing a discussion of the T E andEE likelihood blocks to Sect. 3.2. The main differences betweenthe 2013 and 2015 analyses are as follows.

(1) There have been a number of changes to the low-levelPlanck data processing, as discussed in Planck Collaboration II(2016) and Planck Collaboration VII (2016). These include:changes to the filtering applied to remove “4-K” cooler linesfrom the time-ordered data (TOD); changes to the deglitchingalgorithm used to correct the TOD for cosmic ray hits; improvedabsolute calibration based on the spacecraft orbital dipole andmore accurate models of the beams, accounting for the interme-diate and far sidelobes. These revisions largely eliminate the cal-ibration difference between Planck-2013 and WMAP reported inPCP13 and Planck Collaboration XXXI (2014), leading to up-ward shifts of the HFI and LFI Planck power spectra of approx-imately 2.0 % and 1.7 %, respectively. In addition, the mapmak-ing used for 2015 data processing utilizes “polarization destrip-ing” for the polarized HFI detectors (Planck Collaboration VIII2016).

(2) The 2013 papers used WMAP polarization measurements(Bennett et al. 2013) at multipoles ` ≤ 23 to constrain the opticaldepth parameter τ; this likelihood was denoted “WP” in the 2013papers. In the 2015 analysis, the WMAP polarization likelihoodis replaced by a Planck polarization likelihood constructed from

5


Table 1. Parameters of the base ΛCDM cosmology (as defined in PCP13) determined from the publicly released nominal-missionCamSpecDetSet likelihood [2013N(DS)] and the 2013 full-mission CamSpecDetSet and cross-yearly (Y1×Y2) likelihoods with theextended sky coverage [2013F(DS) and 2013F(CY)]. These three likelihoods are combined with the WMAP polarization likelihoodto constrain τ. The column labelled 2015F(CHM) lists parameters for a CamSpec cross-half-mission likelihood constructed fromthe 2015 maps using similar sky coverage to the 2013F(CY) likelihood (but greater sky coverage at 217 GHz and different pointsource masks, as discussed in the text). The column labelled 2015F(CHM) (Plik) lists parameters for the Plik cross-half-missionlikelihood that uses identical sky coverage to the CamSpec likelihood. The 2015 temperature likelihoods are combined with thePlanck lowP likelihood to constrain τ. The last two columns list the deviations of the Plik parameters from those of the nominal-mission and the CamSpec 2015(CHM) likelihoods. To help refer to specific columns, we have numbered the first six explicitly. Thehigh-` likelihoods used here include only TT spectra. H0 is given in the usual units of km s−1 Mpc−1.

[1] Parameter [2] 2013N(DS) [3] 2013F(DS) [4] 2013F(CY) [5] 2015F(CHM) [6] 2015F(CHM) (Plik) ([2] − [6])/σ[6] ([5] − [6])/σ[5]

100θMC . . . . . . . . . 1.04131 ± 0.00063 1.04126 ± 0.00047 1.04121 ± 0.00048 1.04094 ± 0.00048 1.04086 ± 0.00048 0.71 0.17Ωbh2 . . . . . . . . . . . 0.02205 ± 0.00028 0.02234 ± 0.00023 0.02230 ± 0.00023 0.02225 ± 0.00023 0.02222 ± 0.00023 −0.61 0.13Ωch2 . . . . . . . . . . . 0.1199 ± 0.0027 0.1189 ± 0.0022 0.1188 ± 0.0022 0.1194 ± 0.0022 0.1199 ± 0.0022 0.00 −0.23H0 . . . . . . . . . . . . 67.3 ± 1.2 67.8 ± 1.0 67.8 ± 1.0 67.48 ± 0.98 67.26 ± 0.98 0.03 0.22ns . . . . . . . . . . . . 0.9603 ± 0.0073 0.9665 ± 0.0062 0.9655 ± 0.0062 0.9682 ± 0.0062 0.9652 ± 0.0062 −0.67 0.48Ωm . . . . . . . . . . . . 0.315 ± 0.017 0.308 ± 0.013 0.308 ± 0.013 0.313 ± 0.013 0.316 ± 0.014 −0.06 −0.23σ8 . . . . . . . . . . . . 0.829 ± 0.012 0.831 ± 0.011 0.828 ± 0.012 0.829 ± 0.015 0.830 ± 0.015 −0.08 −0.07τ . . . . . . . . . . . . . 0.089 ± 0.013 0.096 ± 0.013 0.094 ± 0.013 0.079 ± 0.019 0.078 ± 0.019 0.85 0.05109Ase−2τ . . . . . . . . 1.836 ± 0.013 1.833 ± 0.011 1.831 ± 0.011 1.875 ± 0.014 1.881 ± 0.014 −3.46 −0.42

low-resolution maps of Q and U polarization measured by LFI at70 GHz, foreground cleaned using the LFI 30-GHz and HFI 353-GHz maps as polarized synchrotron and dust templates, respec-tively, as described in Planck Collaboration XI (2016). After acomprehensive analysis of survey-to-survey null tests, we foundpossible low-level residual systematics in Surveys 2 and 4,likely related to the unfavourable alignment of the CMB dipolein those two surveys (for details see Planck Collaboration II2016). We therefore conservatively use only six of the eightLFI 70-GHz full-sky surveys, excluding Surveys 2 and 4, Theforeground-cleaned LFI 70-GHz polarization maps are used over46 % of the sky, together with the temperature map from theCommander component-separation algorithm over 94 % of thesky (see Planck Collaboration IX 2016, for further details), toform a low-` Planck temperature+polarization pixel-based like-lihood that extends up to multipole ` = 29. Use of the polariza-tion information in this likelihood is denoted as “lowP” in thispaper The optical depth inferred from the lowP likelihood com-bined with the Planck TT likelihood is typically τ ≈ 0.07, andis about 1σ lower than the typical values of τ ≈ 0.09 inferredfrom the WMAP polarization likelihood (see Sect. 3.4) used inthe 2013 papers. As discussed in Sect. 3.4 (and in more detailin Planck Collaboration XI 2016) the LFI 70-GHz and WMAPpolarization maps are consistent when both are cleaned with theHFI 353-GHz polarization maps.7

(3) In the 2013 papers, the Planck temperature likelihood wasa hybrid: over the multipole range `= 2–49, the likelihoodwas based on the Commander algorithm applied to 87 % of

7Throughout this paper, we adopt the following labels for likeli-hoods: (i) Planck TT denotes the combination of the TT likelihood atmultipoles ` ≥ 30 and a low-` temperature-only likelihood based onthe CMB map recovered with Commander; (ii) Planck TT+lowP fur-ther includes the Planck polarization data in the low-` likelihood, as de-scribed in the main text; (iii) labels such as Planck TE+lowP denote theT E likelihood at ` ≥ 30 plus the polarization-only component of themap-based low-` Planck likelihood; and (iv) Planck TT,TE,EE+lowPdenotes the combination of the likelihood at ` ≥ 30 using TT , T E,and EE spectra and the low-` temperature+polarization likelihood. Wemake occasional use of combinations of the polarization likelihoods at` ≥ 30 and the temperature+polarization data at low-`, which we denotewith labels such as Planck TE+lowT,P.

the sky computed using a Blackwell-Rao estimatorl the likeli-hood at higher multipoles (`=50–2500) was constructed fromcross-spectra over the frequency range 100–217 GHz using theCamSpec software (Planck Collaboration XV 2014), which isbased on the methodology developed in Efstathiou (2004) andEfstathiou (2006). At each of the Planck HFI frequencies, thesky is observed by a number of detectors. For example, at217 GHz the sky is observed by four unpolarized spider-webbolometers (SWBs) and eight polarization sensitive bolometers(PSBs). The TOD from the 12 bolometers can be combined toproduce a single map at 217 GHz for any given period of time.Thus, we can produce 217-GHz maps for individual sky surveys(denoted S1, S2, S3, etc.), or by year (Y1, Y2), or split by half-mission (HM1, HM2). We can also produce a temperature mapfrom each SWB and a temperature and polarization map fromquadruplets of PSBs. For example, at 217 GHz we produce fourtemperature and two temperature+polarization maps. We referto these maps as detectors-set maps (or “DetSets” for short);note that the DetSet maps can also be produced for any arbitrarytime period. The high multipole likelihood used in the 2013 pa-pers was computed by cross-correlating HFI DetSet maps forthe “nominal” Planck mission extending over 15.5 months.8 Forthe 2015 papers we use the full-mission Planck data, extendingover 29 months for the HFI and 48 months for the LFI. In thePlanck 2015 analysis, we have produced cross-year and cross-half-mission likelihoods in addition to a DetSet likelihood. Thebaseline 2015 Planck temperature-polarization likelihood is alsoa hybrid, matching the high-multipole likelihood at ` = 30 to thePlanck pixel-based likelihood at lower multipoles.

(4) The sky coverage used in the 2013 CamSpec likelihood wasintentionally conservative, retaining effectively 49 % of the skyat 100 GHz and 31 % of the sky at 143 and 217 GHz.9 This wasdone to ensure that on the first exposure of Planck cosmologicalresults to the community, corrections for Galactic dust emissionwere demonstrably small and had negligible impact on cosmo-

8Although we analysed a Planck full-mission temperature likeli-hood extensively, prior to the release of the 2013 papers.

9These quantities are explicitly the apodized effective f effsky, calcu-

lated as the average of the square of the apodized mask values (seeEq. 10).

6


logical parameters. In the 2015 analysis we make more aggres-sive use of the sky at each of these frequencies. We have alsotuned the point-source masks to each frequency, rather than us-ing a single point-source mask constructed from the union ofthe point source catalogues at 100, 143, 217, and 353 GHz. Thisresults in many fewer point source holes in the 2015 analysiscompared to the 2013 analysis.

(5) Most of the results in this paper are derived from a revisedPlik likelihood, based on cross-half-mission spectra. The Pliklikelihood has been modified since 2013 so that it is now similarto the CamSpec likelihood used in PCP13. Both likelihoods usesimilar approximations to compute the covariance matrices. Themain difference is in the treatment of Galactic dust correctionsin the analysis of the polarization spectra. The two likelihoodshave been written independently and give similar (but not iden-tical) results, as discussed further below. The Plik likelihoodis discussed in Planck Collaboration XI (2016). The CamSpeclikelihood is discussed in a separate paper (Efstathiou et al. inpreparation).

(6) We have made minor changes to the foreground modellingand to the priors on some of the foreground parameters, as dis-cussed in Sect. 2.3 and Planck Collaboration XI (2016).

Given these changes to data processing, mission length, skycoverage, etc., it is reasonable to ask whether the base ΛCDMparameters have changed significantly compared to the 2013numbers. In fact, the parameter shifts are relatively small. Thesituation is summarized in Table 1. The second column of thistable lists the Planck+WP parameters, as given in table 5 ofPCP13. Since these numbers are based on the 2013 processing ofthe nominal mission and computed via a DetSet CamSpec likeli-hood, the column is labelled 2013N(DS). We now make a num-ber of specific remarks about these comparisons.

(1) 4-K cooler line systematics. After the submission of PCP13we found strong evidence that a residual in the 217×217 DetSetspectrum at ` ≈ 1800 was a systematic caused by electromag-netic interference between the Joule-Thomson 4-K cooler elec-tronics and the bolometer readout electronics. This interferenceleads to a set of time-variable narrow lines in the power spec-trum of the TOD. The data processing pipelines apply a filter toremove these lines; however, the filtering failed to reduce theirimpact on the power spectra to negligible levels. Incomplete re-moval of the 4-K cooler lines affects primarily the 217 × 217PSB×PSB cross-spectrum in Survey 1. The presence of this sys-tematic was reported in the revised versions of 2013 Planckpapers. Using simulations and also comparison with the 2013full-mission likelihood (in which the 217 × 217 power spec-trum “dip” is strongly diluted by the additional sky surveys)we assessed that the 4-K line systematic was causing shiftsin cosmological parameters of less than 0.5σ.10 Column 3 inTable 1 lists the DetSet parameters for the full-mission 2013data. This full-mission likelihood uses more extensive sky cov-erage than the nominal mission likelihood (effectively 39 % ofsky at 217 GHz, 55 % of sky at 143 GHz, and 63 % of sky at

10The revised version of PCP13 also reported an error in the orderingof the beam-transfer functions applied to some of the 2013 217 × 217DetSet cross-spectra, leading to an offset of a few (µK)2 in the coadded217 × 217 spectrum. As discussed in PCP13, this offset is largely ab-sorbed by the foreground model and has negligible impact on the 2013cosmological parameters.

100 GHz); otherwise the methodology and foreground model areidentical to the CamSpec likelihood described in PPL13. The pa-rameter shifts are relatively small and consistent with the im-provement in signal-to-noise of the full-mission spectra and thesystematic shifts caused by the 217×217 dip in the nominal mis-sion (for example, raising H0 and ns, as discussed in appendix C4of PCP13).

(2) DetSets versus cross-surveys. In a reanalysis of the pub-licly released Planck maps, Spergel et al. (2015) constructedcross-survey (S1 × S2) likelihoods and found cosmological pa-rameters for the base ΛCDM model that were close to (withinapproximately 1σ) the nominal mission parameters listed inTable 1. The Spergel et al. (2015) analysis differs substantiallyin sky coverage and foreground modelling compared to the 2013Planck analysis and so it is encouraging that they find no majordifferences with the results presented by the Planck collabora-tion. On the other hand, they did not identify the reasons for theroughly 1σ parameter shifts. They argue that foreground mod-elling and the `= 1800 dip in the 217 × 217 DetSet spectrumcan contribute towards some of the differences but cannot pro-duce 1σ shifts, in agreement with the conclusions of PCP13.The 2013F(DS) likelihood disfavours the Spergel et al. (2015)cosmology (with parameters listed in their table 3) by ∆χ2 = 11,i.e., by about 2σ, and almost all of the ∆χ2 is contributed bythe multipole range 1000–1500, so the parameter shifts are notdriven by cotemporal systematics resulting in correlated noisebiases at high multipoles. However, as discussed in PPL13 andPlanck Collaboration XI (2016), low-level correlated noise inthe DetSet spectra affects all HFI channels at high multipoleswhere the spectra are noise dominated. The impact of this corre-lated noise on cosmological parameters is relatively small. Thisis illustrated by column 4 of Table 1 (labelled “2013F(CY)”),which lists the parameters of a 2013 CamSpec cross-year like-lihood using the same sky coverage and foreground model asthe DetSet likelihood used for column 3. The parameters fromthese two likelihoods are in good agreement (better than 0.2σ),illustrating that cotemporal systematics in the DetSets are at suf-ficiently low levels that there is very little effect on cosmolog-ical parameters. Nevertheless, in the 2015 likelihood analysiswe apply corrections for correlated noise to the DetSet cross-spectra, as discussed in Planck Collaboration XI (2016), andtypically find agreement in cosmological parameters betweenDetSet, cross-year, and cross-half-mission likelihoods to betterthan 0.5σ accuracy for a fixed likelihood code (and to betterthan 0.2σ accuracy for base ΛCDM).

(3) 2015 versus 2013 processing. Column 5 (labelled“2015F(CHM)”) lists the parameters computed from theCamSpec cross-half-mission likelihood using the HFI 2015 datawith revised absolute calibration and beam-transfer functions.We also replace the WP likelihood of the 2013 analysis withthe Planck lowP likelihood. The 2015F(CHM) likelihood usesslightly more sky coverage (60 %) at 217 GHz, compared tothe 2013F(CY) likelihood and also uses revised point sourcemasks. Despite these changes, the base ΛCDM parameters de-rived from the 2015 CamSpec likelihood are within ≈ 0.4σ ofthe 2013F(CY) parameters, with the exception of θMC, which islower by 0.67σ, τ, which is lower by 1σ, and Ase−2τ, which ishigher by about 4σ . The change in τ simply reflects the prefer-ence for a lower value of τ from the Planck LFI polarization datacompared to the WMAP polarization likelihood in the form de-livered by the WMAP team (see Sect. 3.4 for further discussion).

7


0

1000

2000

3000

4000

5000

6000

DTT

`[µ

K2]

30 500 1000 1500 2000 2500`

-60-3003060

∆DTT

`

2 10-600-300

0300600

Fig. 1. Planck 2015 temperature power spectrum. At multipoles ` ≥ 30 we show the maximum likelihood frequency-averagedtemperature spectrum computed from the Plik cross-half-mission likelihood, with foreground and other nuisance parameters de-termined from the MCMC analysis of the base ΛCDM cosmology. In the multipole range 2 ≤ ` ≤ 29, we plot the power spectrumestimates from the Commander component-separation algorithm, computed over 94 % of the sky. The best-fit base ΛCDM theoreti-cal spectrum fitted to the Planck TT+lowP likelihood is plotted in the upper panel. Residuals with respect to this model are shownin the lower panel. The error bars show ±1σ uncertainties.

The large upward shift in Ase−2τ reflects the change in the abso-lute calibration of the HFI. As noted in Sect. 2.3, the 2013 analy-sis did not propagate an error on the Planck absolute calibrationthrough to cosmological parameters. Coincidentally, the changesto the absolute calibration compensate for the downward changein τ and variations in the other cosmological parameters to keepthe parameter σ8 largely unchanged from the 2013 value. Thiswill be important when we come to discuss possible tensionsbetween the amplitude of the matter fluctuations at low redshiftestimated from various astrophysical data sets and the PlanckCMB values for the base ΛCDM cosmology (see Sect. 5.6).

(4) Likelihoods. Constructing a high-multipole likelihood forPlanck, particularly with T E and EE spectra, is complicatedand difficult to check at the sub-σ level against numericalsimulations because the simulations cannot model the fore-grounds, noise properties, and low-level data processing ofthe real Planck data to sufficiently high accuracy. Within thePlanck collaboration, we have tested the sensitivity of the re-sults to the likelihood methodology by developing several in-dependent analysis pipelines. Some of these are described inPlanck Collaboration XI (2016). The most highly developed of

them are the CamSpec and revised Plik pipelines. For the 2015Planck papers, the Plik pipeline was chosen as the baseline.Column 6 of Table 1 lists the cosmological parameters for baseΛCDM determined from the Plik cross-half-mission likeli-hood, together with the lowP likelihood, applied to the 2015full-mission data. The sky coverage used in this likelihood isidentical to that used for the CamSpec 2015F(CHM) likelihood.However, the two likelihoods differ in the modelling of instru-mental noise, Galactic dust, treatment of relative calibrations,and multipole limits applied to each spectrum.

As summarized in column 8 of Table 1, the Plik andCamSpec parameters agree to within 0.2σ, except for ns, whichdiffers by nearly 0.5σ. The difference in ns is perhaps not sur-prising, since this parameter is sensitive to small differences inthe foreground modelling. Differences in ns between Plik andCamSpec are systematic and persist throughout the grid of ex-tended ΛCDM models discussed in Sect. 6. We emphasize thatthe CamSpec and Plik likelihoods have been written indepen-dently, though they are based on the same theoretical framework.None of the conclusions in this paper (including those based onthe full “TT,TE,EE” likelihoods) would differ in any substantiveway had we chosen to use the CamSpec likelihood in place ofPlik. The overall shifts of parameters between the Plik 2015

8


likelihood and the published 2013 nominal mission parametersare summarized in column 7 of Table 1. These shifts are within0.7σ except for the parameters τ and Ase−2τ, which are sensitiveto the low-multipole polarization likelihood and absolute cali-bration.

In summary, the Planck 2013 cosmological parameters werepulled slightly towards lower H0 and ns by the ` ≈ 1800 4-K linesystematic in the 217 × 217 cross-spectrum, but the net effect ofthis systematic is relatively small, leading to shifts of 0.5σ orless in cosmological parameters. Changes to the low-level dataprocessing, beams, sky coverage, etc., as well as the likelihoodcode also produce shifts of typically 0.5σ or less. The combinedeffect of these changes is to introduce parameter shifts relative toPCP13 of less than 0.7σ, with the exception of τ and Ase−2τ. Themain scientific conclusions of PCP13 are therefore consistentwith the 2015 Planck analysis.

Parameters for the base ΛCDM cosmology derived fromfull-mission DetSet, cross-year, or cross-half-mission spectra arein extremely good agreement, demonstrating that residual (i.e.,uncorrected) cotemporal systematics are at low levels. This isalso true for the extensions of the ΛCDM model discussed inSect. 6. It is therefore worth explaining why we have adoptedthe cross-half-mission likelihood as the baseline for this andother 2015 Planck papers. The cross-half-mission likelihood haslower signal-to-noise than the full-mission DetSet likelihood;however, the errors on the cosmological parameters from thetwo likelihoods are almost identical, as can be seen from theentries in Table 1. This is also true for extended ΛCDM models.However, for more complicated tests, such as searches for lo-calized features in the power spectra (Planck Collaboration XX2016), residual 4-K line systematic effects and residual uncor-rected correlated noise at high multipoles in the DetSet likeli-hood can produce results suggestive of new physics (though notat a high significance level). We have therefore decided to adoptthe cross-half-mission likelihood as the baseline for the 2015analysis, sacrificing some signal-to-noise in favour of reducedsystematics. For almost all of the models considered in this pa-per, the Planck results are limited by small systematics of vari-ous types, including systematic errors in modelling foregrounds,rather than by signal-to-noise.

The foreground-subtracted, frequency-averaged, cross-half-mission spectrum is plotted in Fig. 1, together with theCommander power spectrum at multipoles ` ≤ 29. The highmultipole spectrum plotted in this figure is an approximate max-imum likelihood solution based on equations (A24) and (A25) ofPPL13, with the foregrounds and nuisance parameters for eachspectrum fixed to the best-fit values of the base ΛCDM solu-tion. Note that a different way of solving for the Planck CMBspectrum, by marginalizing over foreground and nuisance pa-rameters, is presented in Sect. 4. The best-fit base ΛCDM modelis plotted in the upper panel, while residuals with respect to thismodel are plotted in the lower panel. In this plot, there are onlyfour bandpowers at ` ≥ 30 that differ from the best-fit modelby more than 2σ. These are: `= 434 (−2.0σ); `= 465 (2.5σ);`= 1214 (−2.5σ); and `= 1455 (−2.1σ). The χ2 of the coaddedTT spectrum plotted in Fig. 1 relative to the best-fit base ΛCDMmodel is 2547 for 2479 degrees of freedom (30 ≤ ` ≤ 2500),which is a 0.96σ fluctuation (PTE = 16.8 %). These numbersconfirm the extremely good fit of the base ΛCDM cosmologyto the Planck TT data at high multipoles. The consistency ofthe Planck polarization spectra with base ΛCDM is discussed inSect. 3.3.

PCP13 noted some mild internal tensions within the Planckdata, for example, the preference of the phenomenological lens-

ing parameter AL (see Sect. 5.1) towards values greater thanunity and a preference for a negative running of the scalar spec-tral index (see Sect. 6.2.2). These tensions were partly causedby the poor fit of base ΛCDM model to the temperature spec-trum at multipoles below about 50. As noted by the WMAPteam (Hinshaw et al. 2003), the temperature spectrum has a lowquadrupole amplitude and a glitch in the multipole range 20 <∼` <∼ 30. These features can be seen in the Planck 2015 spectrumof Fig. 1. They have a similar (though slightly reduced) effect oncosmological parameters to those described in PCP13.

3.2. 545-GHz-cleaned spectra

As discussed in PCP13, unresolved extragalactic foregrounds(principally Poisson point sources and the clustered componentof the CIB) contribute to the Planck TT spectra at high mul-tipoles. The approach to modelling these foreground contribu-tions in PCP13 is similar to that used by the ACT and SPTteams (Reichardt et al. 2012; Dunkley et al. 2013) in that theforegrounds are modelled by a set of physically motivated powerspectrum template shapes with an associated set of adjustablenuisance parameters. This approach has been adopted as thebaseline for the Planck 2015 analysis. The foreground model hasbeen adjusted for this new analysis, in relatively minor ways,as summarized in Sect. 2.3 and described in further detail inPlanck Collaboration XII (2016). Galactic dust emission alsocontributes to the temperature and polarization power spectraand must be subtracted from the spectra used to form the Plancklikelihood. Unlike the extragalactic foregrounds, Galactic dustemission is anisotropic and so its impact can be reduced by ap-propriate masking of the sky. In PCP13, we intentionally adoptedconservative masks, tuned for each of the frequencies used toform the likelihood, to keep dust emission at low levels. Theresults in PCP13 were therefore insensitive to the modelling ofresidual dust contamination.

In the 2015 analysis, we have extended the sky coverage ateach of 100, 143, and 217 GHz, and so in addition to testing theaccuracy of the extragalactic foreground model, it is importantto test the accuracy of the Galactic dust model. As describedin PPL13 and Planck Collaboration XII (2016) the Galactic dusttemplates used in the CamSpec and Plik likelihoods are derivedby fitting the 545-GHz mask-differenced power spectra. Maskdifferencing isolates the anisotropic contribution of Galactic dustfrom the isotropic extragalactic components. For the extendedsky coverage used in the 2015 likelihoods, the Galactic dustcontributions are a significant fraction of the extragalactic fore-ground contribution in the 217 × 217 temperature spectrum athigh multipoles, as illustrated in Fig. 2. Galactic dust dominatesover all other foregrounds at multipoles ` <∼ 500 at HFI frequen-cies.

A simple and direct test of the parametric foreground mod-elling used in the CamSpec and Plik likelihoods is to compareresults with a completely different approach in which the low-frequency maps are “cleaned” using higher frequency maps asforeground templates (see, e.g., Lueker et al. 2010). In a similarapproach to Spergel et al. (2015), we can form cleaned maps atlower frequencies ν by subtracting a 545-GHz map as a template,

MTνclean = (1 + αTν )MTν − αTν MTνt , (6)

where νt is the frequency of the template map MTνt and αTν isthe cleaning coefficient. Since the maps have different beams,

9


Fig. 2. Residual plots illustrating the accuracy of the foreground modelling. The blue points in the upper panels show the CamSpec2015(CHM) spectra after subtraction of the best-fit ΛCDM spectrum. The residuals in the upper panel should be accurately de-scribed by the foreground model. Major foreground components are shown by the solid lines, colour coded as follows: total fore-ground spectrum (red); Poisson point sources (orange); clustered CIB (blue); thermal SZ (green); and Galactic dust (purple). Minorforeground components are shown by the dotted lines, colour-coded as follows: kinetic SZ (green); and tSZ×CIB cross-correlation(purple). The red points in the upper panels show the 545-GHz-cleaned spectra (minus the best-fit CMB as subtracted from theuncleaned spectra) that are fitted to a power-law residual foreground model, as discussed in the text. The lower panels show thespectra after subtraction of the best-fit foreground models. These agree to within a few (µK)2. The χ2 values of the residuals of theblue points, and the number of bandpowers, are listed in the lower panels.

the subtraction is actually done in the power spectrum domain:

CTν1 Tν2 clean = (1 + αTν1 )(1 + αTν2 )CTν1 Tν2

−(1 + αTν1 )αTν2 CTν2 Tνt

−(1 + αTν2 )αTν1 CTν1 Tνt + αTν1αTν2 CTνt Tνt , (7)

where CTν1 Tν2 etc. are the mask-deconvolved beam-correctedpower spectra. The coefficients αTνi are determined by minimiz-ing

`max∑`=`min

`max∑`′=`min

CTνi Tνi clean`

(M

Tνi Tνi``′

)−1C

Tνi Tνi clean`′

, (8)

where MTνi Tνi is the covariance matrix of the estimates CTνi Tνi .We choose `min = 100 and `max = 500 and compute the spectra inEq. (7) by cross-correlating half-mission maps on the 60 % maskused to compute the 217× 217 spectrum. The resulting cleaningcoefficients are αT

143 = 0.00194 and αT217 = 0.00765; note that

all of the input maps are in units of thermodynamic tempera-ture. The cleaning coefficients are therefore optimized to removeGalactic dust at low multipoles, though by using 545 GHz as adust template we find that the cleaning coefficients are almostconstant over the multipole range 50–2500. We note, however,that this is not true if the 353- and 857-GHz maps are used asdust templates, as discussed in Efstathiou et al. (in preparation).

The 545-GHz-cleaned spectra are shown by the red pointsin Fig. 2 and can be compared directly to the “uncleaned” spec-tra used in the CamSpec likelihood (upper panels). As can beseen, Galactic dust emission is removed to high accuracy and theresidual foreground contribution at high multipoles is stronglysuppressed in the 217×217 and 143×217 spectra. Nevertheless,there remains small foreground contributions at high multipoles,which we model heuristically as power laws,

D` = A(

`

1500

)ε, (9)

with free amplitudes A and spectral indices ε. We construct an-other CamSpec cross-half-mission likelihood using exactly thesame sky masks as the 2015F(CHM) likelihood, but using 545-GHz-cleaned 217 × 217, 143 × 217, and 143 × 143 spectra. Wethen use the simple model of Eq. (9) in the likelihood to removeresidual unresolved foregrounds at high multipoles for each fre-quency combination. We do not clean the 100×100 spectrum andso for this spectrum we use the standard parametric foregroundmodel in the likelihood. The lower panels in Fig. 2 show theresiduals with respect to the best-fit base ΛCDM model and fore-ground solution for the uncleaned CamSpec spectra (blue points)and for the 545-GHz-cleaned spectra (red points). These resid-uals are almost identical, despite the very different approachesto Galactic dust removal and foreground modelling. The cosmo-logical parameters from these two likelihoods are also in verygood agreement, typically to better than 0.1σ, with the excep-tion of ns, which is lower in the cleaned likelihood by 0.26σ. Itis not surprising, given the heuristic nature of the model (Eq. 9),that ns shows the largest shift. We can also remove the 100×100spectrum from the likelihood entirely, with very little impact oncosmological parameters.

Further tests of map-based cleaning are presented inPlanck Collaboration XI (2016), which additionally describesanother independently written power-spectrum analysis pipeline(MSPEC) tuned to map-cleaned cross-spectrum analysis and us-ing a more complex model for fitting residual foregroundsthan the heuristic model of Eq. (9). Planck Collaboration XI(2016) also describes power spectrum analysis and cosmologi-cal parameters derived from component-separated Planck maps.However, the simple demonstration presented in this sectionshows that the details of modelling residual dust contaminationand other foregrounds are under control in the 2015 Planck like-lihood. A further strong argument that our TT results are insen-sitive to foreground modelling is presented in the next section,which compares the cosmological parameters derived from the

10


TT , T E, and EE likelihoods. Unresolved foregrounds at highmultipoles are completely negligible in the polarization spec-tra and so the consistency of the parameters, particularly fromthe T E spectrum (which has higher signal-to-noise than the EEspectrum) provides an additional cross-check of the TT results.

Finally, one can ask why we have not chosen to use a 545-GHz-cleaned likelihood as the baseline for the 2015 Planck pa-rameter analysis. Firstly, it would not make any difference tothe results of this paper had we chosen to do so. Secondly, wefeel that the parametric foreground model used in the baselinelikelihood has a sounder physical basis. This allows us to linkthe amplitudes of the unresolved foregrounds across the variousPlanck frequencies with the results from other ways of studyingforegrounds, including the higher resolution CMB experimentsdescribed in Sect. 4.

3.3. The 2015 Planck temperature and polarization spectraand likelihood

The coadded 2015 Planck temperature spectrum was introducedin Fig. 1. In this section, we present additional details and con-sistency checks of the temperature likelihood and describe thefull mission Planck T E and EE spectra and likelihood; pre-liminary Planck T E and EE spectra were presented in PCP13.We then discuss the consistency of the cosmological parame-ters for base ΛCDM measured separately from the TT , T E,and EE spectra. For the most part, the discussion given in thissection is specific to the Plik likelihood, which is used as thebaseline in this paper. A more complete discussion of the Plikand other likelihoods developed by the Planck team is given inPlanck Collaboration XI (2016).

3.3.1. Temperature spectra and likelihood

(1) Temperature masks. As in the 2013 analysis, the high-multipole TT likelihood uses the 100×100 , 143×143, 217×217,and 143 × 217 spectra. However, in contrast to the 2013 anal-ysis, which used conservative sky masks to reduce the effectsof Galactic dust emission, we make more aggressive use of skyin the 2015 analysis. The 2015 analysis retains 80 %, 70 %,and 60 % of sky at 100 GHz, 143 GHz, and 217 GHz, respec-tively, before apodization. We also apply apodized point sourcemasks to remove compact sources with a signal-to-noise thresh-old > 5 at each frequency (see Planck Collaboration XXVI 2016for a description of the Planck Catalogue of Compact Sources).Apodized masks are also applied to remove extended objects,and regions of high CO emission were masked at 100 GHz and217 GHz (see Planck Collaboration X 2016). As an estimate ofthe effective sky area, we compute the following sum over pix-els:

f effsky =

14π

∑w2

i Ωi, (10)

where wi is the weight of the apodized mask and Ωi is the area ofpixel i. All input maps are at HEALpix (Gorski et al. 2005) res-olution Nside = 2048. Eq. (10) gives f eff

sky = 66.3 % at 100 GHz,57.4 % at 143 GHz, and 47.1 % at 217 GHz.

(2) Galactic dust templates. With the increased sky coverageused in the 2015 analysis, we take a slightly different approachto subtracting Galactic dust emission to that described in PPL13and PCP13. The shape of the Galactic dust template is deter-mined from mask-differenced power spectra estimated from the545-GHz maps. The mask differencing removes the isotropic

contribution from the CIB and point sources. The resulting dusttemplate has a similar shape to the template used in the 2013analysis, with power-law behaviourDdust

` ∝ `−0.63 at high multi-poles, but with a “bump” at ` ≈ 200 (as shown in Fig. 2). The ab-solute amplitude of the dust templates at 100, 143, and 217 GHzis determined by cross-correlating the temperature maps at thesefrequencies with the 545-GHz maps (with minor corrections forthe CIB and point source contributions). This allows us to gen-erate priors on the dust template amplitudes, which are treatedas additional nuisance parameters when running MCMC chains(unlike the 2013 analysis, in which we fixed the amplitudes ofthe dust templates). The actual priors used in the Plik likelihoodare Gaussians on Ddust

`=200 with the following means and disper-sions: (7 ± 2) µK2 for the 100 × 100 spectrum; (9 ± 2) µK2 for143 × 143; (21 ± 8.5) µK2 for 143 × 217; and (80 ± 20) µK2 for217 × 217. The MCMC solutions show small movements of thebest-fit dust template amplitudes, but always within statisticallyacceptable ranges given the priors.

(3) Likelihood approximation and covariance matrices. Theapproximation to the likelihood function follows the methodol-ogy described in PPL13 and is based on a Gaussian likelihoodassuming a fiducial theoretical power spectrum (a fit to Plik TTwith prior τ = 0.07 ± 0.02). We have also included a numberof small refinements to the covariance matrices. Foregrounds,including Galactic dust, are added to the fiducial theoreticalpower spectrum, so that the additional small variance associatedwith foregrounds is included, along with cosmic variance of theCMB, under the assumption that the foregrounds are Gaussianrandom fields. The 2013 analysis did not include corrections tothe covariance matrices arising from leakage of low-multipolepower to high multipoles via the point source holes; these canintroduce errors in the covariance matrices of a few percent at` ≈ 300, corresponding approximately to the first peak of theCMB spectrum. In the 2015 analysis we apply corrections to thefiducial theoretical power spectrum, based on Monte Carlo sim-ulations, to correct for this effect. We also apply Monte Carlobased corrections to the analytic covariance matrices at multi-poles ` ≤ 50, where the analytic approximations begin to be-come inaccurate even for large effective sky areas (see Efstathiou2004). Finally, we add the uncertainties on the beam shapes tothe covariance matrix following the methodology described inPPL13. The Planck beams are much more accurately character-ized in the 2015 analysis, and so the beam corrections to thecovariance matrices are extremely small. The refinements to thecovariance matrices described in this paragraph are all relativelyminor and have little impact on cosmological parameters.

(4) Binning. The baseline Plik likelihood uses binned tempera-ture and polarization spectra. This is done because all frequencycombinations of the T E and EE spectra are used in the Pliklikelihood, leading to a large data vector of length 22 865 if thespectra are retained multipole-by-multipole. The baseline Pliklikelihood reduces the size of the data vector by binning thesespectra. The spectra are binned into bins of width ∆` = 5 for30 ≤ ` ≤ 99, ∆` = 9 for 100 ≤ ` ≤ 1503, ∆` = 17 for1504 ≤ ` ≤ 2013 and ∆` = 33 for 2014 ≤ ` ≤ 2508, witha weighting of C` proportional to `(` + 1) over the bin widths.The bins span an odd number of multipoles, since for approxi-mately azimuthal masks we expect a nearly symmetrical correla-tion function around the central multipole. The binning does notaffect the determination of cosmological parameters in ΛCDM-

11


Table 2. Goodness-of-fit tests for the 2015 Planck temperature and polarization spectra. ∆χ2 = χ2 − Ndof is the difference fromthe mean assuming that the best-fit base ΛCDM model (fitted to Planck TT+lowP) is correct and Ndof is the number of degrees offreedom (set equal to the number of multipoles). The sixth column expresses ∆χ2 in units of the expected dispersion,

√2Ndof , and

the last column lists the probability to exceed (PTE) the tabulated value of χ2.

Likelihood Frequency Multipole range χ2 χ2/Ndof Ndof ∆χ2/√

2Ndof PTE [%]

TT 100×100 30–1197 1234.37 1.06 1168 1.37 8.66143×143 30–1996 2034.45 1.03 1967 1.08 14.14143×217 30–2508 2566.74 1.04 2479 1.25 10.73217×217 30–2508 2549.66 1.03 2479 1.00 15.78Combined 30–2508 2546.67 1.03 2479 0.96 16.81

T E 100×100 30– 999 1088.78 1.12 970 2.70 0.45100×143 30– 999 1032.84 1.06 970 1.43 7.90100×217 505– 999 526.56 1.06 495 1.00 15.78143×143 30–1996 2028.43 1.03 1967 0.98 16.35143×217 505–1996 1606.25 1.08 1492 2.09 2.01217×217 505–1996 1431.52 0.96 1492 −1.11 86.66Combined 30–1996 2046.11 1.04 1967 1.26 10.47

EE 100×100 30– 999 1027.89 1.06 970 1.31 9.61100×143 30– 999 1048.22 1.08 970 1.78 4.05100×217 505– 999 479.72 0.97 495 −0.49 68.06143×143 30–1996 2000.90 1.02 1967 0.54 29.18143×217 505–1996 1431.16 0.96 1492 −1.11 86.80217×217 505–1996 1409.58 0.94 1492 −1.51 93.64Combined 30–1996 1986.95 1.01 1967 0.32 37.16

type models (which have smooth power spectra), but signifi-cantly reduces the size of the joint TT,TE,EE covariance ma-trix, speeding up the computation of the likelihood. However, forsome specific purposes, e.g., searching for oscillatory features inthe TT spectrum, or testing χ2 statistics, we produce blocks ofthe likelihood multipole-by-multipole.

(5) Goodness of fit. The first five rows of Table 2 list χ2 statis-tics for the TT spectra (multipole-by-multipole) relative to thePlanck best-fit base ΛCDM model and foreground parameters(fitted to Planck TT+lowP). The first four entries list the statis-tics separately for each of the four spectra that form the TTlikelihood and the fifth line (labelled “Combined”) gives theχ2 value for the maximum likelihood TT spectrum plotted inFig. 1. Each of the individual spectra provides an acceptable fitto the base ΛCDM model, as does the frequency-averaged spec-trum plotted in Fig. 1. This demonstrates the excellent consis-tency of the base ΛCDM model across frequencies. More de-tailed consistency checks of the Planck spectra are presentedin Planck Collaboration XI (2016); however, as indicated byTable 2, we find no evidence for any inconsistencies betweenthe foreground-corrected temperature power spectra computedfor different frequency combinations. The temperature spectraare largely signal dominated over the multipole ranges listed inTable 2 and so the χ2 values are insensitive to small errors inthe Planck noise model used in the covariance matrices. As dis-cussed in the next subsection, this is not true for the T E and EEspectra, which are noise dominated over much of the multipolerange.

3.3.2. Polarization spectra and likelihood

In addition to the TT spectra, the 2015 Planck likelihood in-cludes the T E and EE spectra. As discussed in Sect. 3.1, thePlanck 2015 low-multipole polarization analysis is based on theLFI 70-GHz data. Here we discuss the T E and EE spectra that

are used in the high-multipole likelihood, which are computedfrom the HFI data at 100, 143 and 217 GHz. As summarized inPlanck Collaboration XI (2016), there is no evidence for any un-resolved foreground components at high multipoles in the polar-ization spectra. We therefore include all frequency combinationsin computing the T E and EE spectra to maximize the signal-to-noise.11

(1) Masks and dust corrections. At low multipoles (` <∼ 300)polarized Galactic dust emission is significant at all frequen-cies and is subtracted in a similar way to the dust subtractionin temperature, i.e., by including additional nuisance parame-ters quantifying the amplitudes of a power-law dust templatewith a slope constrained to Ddust

` ∝ `−0.40 for both T E and EE(Planck Collaboration Int. XXX 2016). Polarized synchrotronemission (which has been shown to be negligible at 100 GHzand higher frequencies for Planck noise levels, Fuskeland et al.2014) is ignored. Gaussian priors on the polarization dust ampli-tudes are determined by cross-correlating the lower frequencymaps with the 353-GHz polarization maps (the highest fre-quency polarized channel of the HFI) in a similar way to the de-termination of temperature dust priors. We use the temperature-based apodized masks in Q and U at each frequency, retaining70 %, 50 %, and 41 % of the sky at 100, 143, and 217 GHz, re-spectively, after apodization (slightly smaller than the tempera-ture masks at 143 and 217 GHz). However, we do not apply pointsource or CO masks to the Q and U maps. The construction ofthe full TT,TE,EE likelihood is then a straightforward extensionof the TT likelihood using the analytic covariance matrices givenby Efstathiou (2006) and Hamimeche & Lewis (2008).

11In temperature, the 100 × 143 and 100 × 217 spectra are not in-cluded in the likelihood because the temperature spectra are largely sig-nal dominated. These spectra therefore add little new information onthe CMB, but would require additional nuisance parameters to correctfor unresolved foregrounds at high multipoles.

12


-140

-70

0

70

140

DTE

`[µ

K2]

30 500 1000 1500 2000

`

-10

0

10

∆DT

E`

0

20

40

60

80

100

CEE

`[1

0−

5µ

K2]

30 500 1000 1500 2000

`

-4

0

4

∆CEE

`

Fig. 3. Frequency-averaged T E and EE spectra (without fitting for temperature-to-polarization leakage). The theoretical T E andEE spectra plotted in the upper panel of each plot are computed from the Planck TT+lowP best-fit model of Fig. 1. Residuals withrespect to this theoretical model are shown in the lower panel in each plot. The error bars show ±1σ errors. The green lines in thelower panels show the best-fit temperature-to-polarization leakage model of Eqs. (11a) and (11b), fitted separately to the T E andEE spectra.

13


0 500 1000 1500 2000

`

20

15

10

5

0

5

10

15

20∆DTE`

[µK

2]

0 500 1000 1500 2000

`

10

5

0

5

10

∆CEE

`[1

0−

5µK

2]

Fig. 4. Conditionals for the Plik T E and EE spectra, given the TT data computed from the Plik likelihood. The black lines showthe expected T E and EE spectra given the TT data. The shaded areas show the ±1 and ±2σ ranges computed from Eq. (16). Theblue points show the residuals for the measured T E and EE spectra.

Fig. 5. Conditionals for the CamSpec T E and EE spectra, given the TT data computed from the CamSpec likelihood. As in Fig. 4,the shaded areas show ±1 and ±2σ ranges, computed from Eq. (16) and blue points show the residuals for the measured T E andEE spectra.

(2) Polarization spectra and residual systematics. Maximumlikelihood frequency coadded T E and EE spectra are shown inFig. 3. The theoretical curves plotted in these figures are the T Eand EE spectra computed from the best-fit base ΛCDM modelfitted to the temperature spectra (Planck TT+lowP), as plottedin Fig. 1. The lower panels in each figure show the residualswith respect to this model. The theoretical model provides a verygood fit to the T E and EE spectra. Table 2 lists χ2 statistics forthe T E and EE spectra for each frequency combination (withthe T E and ET spectra for each frequency combination coad-ded to form a single T E spectrum). Note that since the T E andEE spectra are noisier than the TT spectra, these values of χ2

are sensitive to the procedure used to estimate Planck noise (seePlanck Collaboration XI 2016 for further details).

Some of these χ2 values are unusually high, for example the100×100 and 143×217 T E spectra and the 100×143 EE spec-trum all have low PTEs. The Planck T E and EE spectra for dif-ferent frequency combinations are not as internally consistent asthe Planck TT spectra. Inter-comparison of the T E and EE spec-tra at different frequencies is much more straightforward thanfor the temperature spectra because unresolved foregrounds are

unimportant in polarization. The high χ2 values listed in Table 2therefore provide clear evidence of residual instrumental system-atics in the T E and EE spectra.

With our present understanding of the Planck polarizationdata, we believe that the dominant source of systematic error inthe polarization spectra is caused by beam mismatch that gener-ates leakage from temperature to polarization (recalling that theHFI polarization maps are generated by differencing signals be-tween quadruplets of polarization sensitive bolometers). In prin-ciple, with accurate knowledge of the beams this leakage couldbe described by effective polarized beam window functions. Forthe 2015 papers, we use the TT beams rather than polarizedbeams, and characterize temperature-to-polarization leakage us-ing a simplified model. The impact of beam mismatch on thepolarization spectra in this model is

∆CT E` = ε`CTT

` , (11a)

∆CEE` = ε2

`CTT` + 2ε`CT E

` , (11b)

where ε` is a polynomial in multipole. As a consequence of thePlanck scanning strategy, pixels are visited approximately everysix months, with a rotation of the focal plane by 180, leading to

14


a weak coupling to beam modes b`m with odd values of m. Thedominant contributions are expected to come from modes withm = 2 and 4, describing the beam ellipticity. We therefore fit thespectra using a fourth-order polynomial,

ε` = a0 + a2`2 + a4`

4, (12)

treating the coefficients a0, a2, and a4 as nuisance parameters inthe MCMC analysis. We have ignored the odd coefficients of thepolynomial, which should be suppressed by our scanning strat-egy. We do however include a constant term in the polynomial toaccount for small deviations of the polarization efficiency fromunity.

The fit is performed separately on the T E and EE spectra. Adifferent polynomial is used for each cross-frequency spectrum.The coadded corrections are shown in the lower panels of Fig. 3.Empirically, we find that temperature-to-polarization leakagesystematics tend to cancel in the coadded spectra. Although thebest-fit leakage corrections to the coadded spectra are small, thecorrections for individual frequency cross-spectra can be up to 3times larger than those shown in Fig. 3. The model of Eqs. (11a)and (11b) is clearly crude, but gives us some idea of the impactof temperature-to-polarization leakage in the coadded spectra.With our present empirical understanding of leakage, we find acorrelation between the polarization spectra that have the high-est expected temperature-to-polarization leakage and those thatdisplay high χ2 in Table 2. However, the characterization of thisleakage is not yet accurate enough to reduce the χ2 values foreach frequency combination to acceptable levels.

As discussed in PCP13, each Planck data release and ac-companying set of papers should be viewed as a snapshot ofthe state of the Planck analysis at the time of the release.For the 2015 release, we have a high level of confidence inthe temperature power spectra. However, we have definite ev-idence for low-level systematics associated with temperature-to-polarization leakage in the polarization spectra. The tests de-scribed above suggest that these are at low levels of a few (µK)2

in D`. However, temperature-to-polarization leakage can intro-duce correlated features in the spectra, as shown by the EE leak-age model plotted in Fig. 3. Until we have a more accurate char-acterization of these systematics, we urge caution in the inter-pretation of features in the T E and EE spectra. For some of the2015 papers, we use the T E and EE spectra, without leakagecorrections. For most of the models considered in this paper, theTT spectra alone provide tight constraints and so we take a con-servative approach and usually quote the TT results. However,as we will see, we find a high level of consistency between thePlanck TT and full Planck TT,T E, EE likelihoods. Some mod-els considered in Sect. 6 are, however, sensitive to the polariza-tion blocks of the likelihood. Examples include constraints onisocurvature modes, dark matter annihilation, and non-standardrecombination histories. Planck 2015 constraints on these mod-els should be viewed as preliminary, pending a more completeanalysis of polarization systematics, which will be presented inthe next series of Planck papers accompanying a third data re-lease.

(3) T E and EE conditionals. Given the best-fit base ΛCDMcosmology and foreground parameters determined from the tem-perature spectra, one can test whether the T E and EE spectra areconsistent with the TT spectra by computing conditional proba-bilities. Writing the data vector as

C = (CTT , CT E , CEE)T = (XT , XP)T, (13)

where the quantities CTT , CT E , and CEE are the maximum like-lihood freqency co-added foreground-corrected spectra. The co-variance matrix of this vector can be partitioned as

M =

MT MT P

MTT P MP

. (14)

The expected value of the polarization vector, given the observedtemperature vector XT is

XcondP = Xtheory

P + MTT P M−1

T (XT − XtheoryT ), (15)

with covariance

ΣP = MP −MTT PM−1

T MT P. (16)

In Eq. (15), XtheoryT and Xtheory

P are the theoretical temperatureand polarization spectra deduced from minimizing the PlanckTT+lowP likelihood. Equations (15) and (16) give the expecta-tion values and distributions of the polarization spectra condi-tional on the observed temperature spectra. These are shown inFig. 4. Almost all of the data points sit within the ±2σ bandsand in the case of the T E spectra, the data points track the fluc-tuations expected from the TT spectra at multipoles ` <∼ 1000.Figure 4 therefore provides an important additional check of theconsistency of the T E and EE spectra with the base ΛCDM cos-mology.

(4) Likelihood implementation. Section 3.1 showed good con-sistency between the independently written CamSpec and Plikcodes in temperature. The methodology used for the temperaturelikelihoods are very similar, but the treatment of the polarizationspectra in the two codes differs substantially. CamSpec uses low-resolution CMB-subtracted 353-GHz polarization maps thresh-olded by P = (Q2 + U2)1/2 to define diffuse Galactic polariza-tion masks. The same apodized polarization mask, with an effec-tive sky fraction f eff

sky = 48.8 % (as defined by Eq. (10)), is usedfor 100-, 143-, and 217-GHz Q and U maps. Since there areno unresolved extragalactic foregrounds detected in the T E andEE spectra, all of the different frequency combinations of T Eand EE spectra are compressed into single T E and EE spectra(weighted by the inverse of the diagonals of the appropriate co-variance matrices), after foreground cleaning using the 353-GHzmaps12 (generalizing the map cleaning technique described inSect. 3.2 to polarization). This allows the construction of a fullTT,T E, EE likelihood with no binning of the spectra and withno additional nuisance parameters in polarization. As noted inSect. 3.1 the consistency of results from the polarization blocksof the CamSpec and Plik likelihoods is not as good as in tem-perature. Cosmological parameters from fits to the T E and EECamSpec and Plik likelihoods can differ by up to about 1.5σ,although no major science conclusions would change had wechosen to use the CamSpec likelihood as the baseline in this pa-per. We will, however, sometimes quote results from CamSpec inaddition to those from Plik to give the reader an indication ofthe uncertainties in polarization associated with different likeli-hood implementations. Figure 5 shows the CamSpec T E and EEresiduals and error ranges conditional on the best-fit base ΛCDMand foreground model fitted to the CamSpec temperature+lowP

12To reduce the impact of noise at 353 GHz, the map-based cleaningof the T E and EE spectra is applied at ` ≤ 300. At higher multipoles,the polarized dust corrections are small and are subtracted as powerlaws fitted to the Galactic dust spectra at lower multipoles.

15


Table 3. Parameters of the base ΛCDM cosmology computed from the 2015 baseline Planck likelihoods, illustrating the consistencyof parameters determined from the temperature and polarization spectra at high multipoles. Column [1] uses the TT spectra at lowand high multipoles and is the same as column [6] of Table 1. Columns [2] and [3] use only the T E and EE spectra at highmultipoles, and only polarization at low multipoles. Column [4] uses the full likelihood. The last column lists the deviations of thecosmological parameters determined from the Planck TT+lowP and Planck TT,TE,EE+lowP likelihoods.

Parameter [1] Planck TT+lowP [2] Planck TE+lowP [3] Planck EE+lowP [4] Planck TT,TE,EE+lowP ([1] − [4])/σ[1]

Ωbh2 . . . . . . . . . . 0.02222 ± 0.00023 0.02228 ± 0.00025 0.0240 ± 0.0013 0.02225 ± 0.00016 −0.1Ωch2 . . . . . . . . . . 0.1197 ± 0.0022 0.1187 ± 0.0021 0.1150+0.0048

−0.0055 0.1198 ± 0.0015 0.0100θMC . . . . . . . . 1.04085 ± 0.00047 1.04094 ± 0.00051 1.03988 ± 0.00094 1.04077 ± 0.00032 0.2τ . . . . . . . . . . . . . 0.078 ± 0.019 0.053 ± 0.019 0.059+0.022

−0.019 0.079 ± 0.017 −0.1ln(1010As) . . . . . . 3.089 ± 0.036 3.031 ± 0.041 3.066+0.046

−0.041 3.094 ± 0.034 −0.1ns . . . . . . . . . . . . 0.9655 ± 0.0062 0.965 ± 0.012 0.973 ± 0.016 0.9645 ± 0.0049 0.2H0 . . . . . . . . . . . 67.31 ± 0.96 67.73 ± 0.92 70.2 ± 3.0 67.27 ± 0.66 0.0Ωm . . . . . . . . . . . 0.315 ± 0.013 0.300 ± 0.012 0.286+0.027

−0.038 0.3156 ± 0.0091 0.0σ8 . . . . . . . . . . . . 0.829 ± 0.014 0.802 ± 0.018 0.796 ± 0.024 0.831 ± 0.013 0.0109Ase−2τ . . . . . . 1.880 ± 0.014 1.865 ± 0.019 1.907 ± 0.027 1.882 ± 0.012 −0.1

likelihood. The residuals in both T E and EE are similar to thosefrom Plik. The main difference can be seen at low multipolesin the EE spectrum, where CamSpec shows a higher dispersion,consistent with the error model, though there are several highpoints at ` ≈ 200 corresponding to the minimum in the EE spec-trum, which may be caused by small errors in the subtractionof polarized Galactic emission using 353 GHz as a foregroundtemplate (and there are also differences in the covariance matri-ces at high multipoles caused by differences in the methods usedin CamSpec and Plik to estimate noise). Generally, cosmolog-ical parameters determined from the CamSpec likelihood havesmaller formal errors than those from Plik because there are nonuisance parameters describing polarized Galactic foregroundsin CamSpec.

3.3.3. Consistency of cosmological parameters from the TT ,T E, and EE spectra

The consistency between parameters of the base ΛCDM modeldetermined from the Plik temperature and polarization spec-tra are summarized in Table 3 and in Fig. 6. As pointed out byZaldarriaga et al. (1997) and Galli et al. (2014), precision mea-surements of the CMB polarization spectra have the potential toconstrain cosmological parameters to higher accuracy than mea-surements of the TT spectra because the acoustic peaks are nar-rower in polarization and unresolved foreground contributions athigh multipoles are much lower in polarization than in temper-ature. The entries in Table 3 show that cosmological parametersthat do not depend strongly on τ are consistent between the TTand T E spectra, to within typically 0.5σ or better. Furthermore,the cosmological parameters derived from the T E spectra havecomparable errors to the TT parameters. None of the conclu-sions in this paper would change in any significant way were weto use the T E parameters in place of the TT parameters. Theconsistency of the cosmological parameters for base ΛCDM be-tween temperature and polarization therefore gives added confi-dence that Planck parameters are insensitive to the specific de-tails of the foreground model that we have used to correct theTT spectra. The EE parameters are also typically within about1σ of the TT parameters, though because the EE spectra fromPlanck are noisier than the TT spectra, the errors on the EE pa-rameters are significantly larger than those from TT . However,both the T E and EE likelihoods give lower values of τ, As andσ8, by over 1σ compared to the TT solutions. Noticee that the

T E and EE entries in Table 3 do not use any information fromthe temperature in the low-multipole likelihood. The tendencyfor higher values of σ8, As, and τ in the Planck TT+lowP solu-tion is driven, in part, by the temperature power spectrum at lowmultipoles.

Columns [4] and [5] of Table 3 compare the parametersof the Planck TT likelihood with the full Planck TT,T E, EElikelihood. These are in agreement, shifting by less than 0.2σ.Although we have emphasized the presence of systematic ef-fects in the Planck polarization spectra, which are not accountedfor in the errors quoted in column [4] of Table 3, the consis-tency of the Planck TT and Planck TT,T E, EE parameters pro-vides strong evidence that residual systematics in the polariza-tion spectra have little impact on the scientific conclusions in thispaper. The consistency of the base ΛCDM parameters from tem-perature and polarization is illustrated graphically in Fig. 6. As arough rule-of-thumb, for base ΛCDM, or extensions to ΛCDMwith spatially flat geometry, using the full Planck TT,T E, EElikelihood produces improvements in cosmological parametersof about the same size as adding BAO to the Planck TT+lowPlikelihood.

3.4. Constraints on the reionization optical depth parameter τ

The reionization optical depth parameter τ provides an importantconstraint on models of early galaxy evolution and star forma-tion. The evolution of the inter-galactic Lyα opacity measured inthe spectra of quasars can be used to set limits on the epoch ofreionization (Gunn & Peterson 1965). The most recent measure-ments suggest that the reionization of the inter-galactic mediumwas largely complete by a redshift z ≈ 6 (Fan et al. 2006). Thesteep decline in the space density of Lyα-emitting galaxies overthe redshift range 6 <∼ z <∼ 8 also implies a low redshift of reion-ization (Choudhury et al. 2015). As a reference, for the Planckparameters listed in Table 3, instantaneous reionization at red-shift z = 7 results in an optical depth of τ = 0.048.

The optical depth τ can also be constrained from observa-tions of the CMB. The WMAP9 results of Bennett et al. (2013)give τ = 0.089 ± 0.014, corresponding to an instantaneous red-shift of reionization zre = 10.6 ± 1.1. The WMAP constraintcomes mainly from the EE spectrum in the multipole range` = 2–6. It has been argued (e.g., Robertson et al. 2013, and ref-erences therein) that the high optical depth reported by WMAPcannot be produced by galaxies seen in deep redshift surveys,

16


0.04 0.08 0.12 0.16

τ

0.0200

0.0225

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2

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ln(1

010A

s)

0.93

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ns

1.038 1.040 1.042

100θMC

0.04

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0.0200 0.0225 0.0250 0.0275

Ωbh20.10 0.11 0.12 0.13

Ωch22.96 3.04 3.12 3.20

ln(1010As)0.93 0.96 0.99 1.02

ns

Planck EE+lowP

Planck TE+lowP

Planck TT+lowP

Planck TT,TE,EE+lowP

Fig. 6. Comparison of the base ΛCDM model parameter constraints from Planck temperature and polarization data.

even assuming high escape fractions for ionizing photons, im-plying additional sources of photoionizing radiation from stillfainter objects. Evidently, it would be useful to have an indepen-dent CMB measurement of τ.

The τ measurement from CMB polarization is difficult be-cause it is a small signal, confined to low multipoles, requiringaccurate control of instrumental systematics and polarized fore-ground emission. As discussed by Komatsu et al. (2009), uncer-tainties in modelling polarized foreground emission are compa-rable to the statistical error in the WMAP τ measurement. Inparticular, at the time of the WMAP9 analysis there was verylittle information available on polarized dust emission. This sit-uation has been partially rectified by the 353-GHz polariza-

tion maps from Planck (Planck Collaboration Int. XXII 2015;Planck Collaboration Int. XXX 2016). In PPL13, we used pre-liminary 353-GHz Planck polarization maps to clean the WMAPKa, Q, and V maps for polarized dust emission, using WMAPK-band as a template for polarized synchrotron emission. Thislowered τ by about 1σ to τ = 0.075 ± 0.013, compared toτ = 0.089 ± 0.013 using the WMAP dust model.13 However,given the preliminary nature of the Planck polarization analysiswe decided for the Planck 2013 papers to use the WMAP polar-ization likelihood, as produced by the WMAP team.

13Neither of these error estimates reflect the true uncertainty in fore-ground removal.

17


0.0216 0.0224 0.0232

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ax

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109As

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Ωm

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ax

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σ8

5 10 15 20

zre

65.0 67.5 70.0 72.5

H0

Planck TT Planck TT,TE,EE Planck TT+lensing Planck TT,TE,EE+lensing Planck TT,TE,EE+lensing+BAO Planck TT+lowP

Fig. 7. Marginalized constraints on parameters of the base ΛCDM model for various data combinations, excluding low-multipolepolarization, compared to the Planck TT+lowP constraints.

In the 2015 papers, we use Planck polarization maps basedon low-resolution LFI 70-GHz maps, excluding Surveys 2 and 4.These maps are foreground-cleaned using the LFI 30-GHz andHFI 353-GHz maps as polarized synchrotron and dust templates,respectively. These cleaned maps form the polarization part(“lowP’ ) of the low-multipole Planck pixel-based likelihood, asdescribed in Planck Collaboration XI (2016). The temperaturepart of this likelihood is provided by the Commander component-separation algorithm. The Planck low-multipole likelihood re-tains 46 % of the sky in polarization and is completely inde-pendent of the WMAP polarization likelihood. In combinationwith the Planck high multipole TT likelihood, the Planck low-multipole likelihood gives τ = 0.078 ± 0.019. This constraintis somewhat higher than the constraint τ = 0.067 ± 0.022derived from the Planck low-multipole likelihood alone (seePlanck Collaboration XI 2016 and also Sect. 5.1.2).

Following the 2013 analysis, we have used the 2015 HFI353-GHz polarization maps as a dust template, together withthe WMAP K-band data as a template for polarized synchrotronemission, to clean the low-resolution WMAP Ka, Q, and V maps(see Planck Collaboration XI 2016 for further details). For thepurpose of cosmological parameter estimation, this data set ismasked using the WMAP P06 mask, which retains 73 % ofthe sky. The noise-weighted combination of the Planck 353-cleaned WMAP polarization maps yields τ = 0.071 ± 0.013when combined with the Planck TT information in the range2 ≤ ` <∼ 2508, consistent with the value of τ obtained fromthe LFI 70-GHz polarization maps. In fact, null tests describedin Planck Collaboration XI (2016) demonstrate that the LFI andWMAP polarization data are statistically consistent. The HFIpolarization maps have higher signal-to-noise than the LFI andcould, in principle, provide a third cross-check. However, at thetime of writing, we are not yet confident that systematics in theHFI maps at low multipoles (` <∼ 20) are at negligible levels. Adiscussion of HFI polarization at low multipoles will thereforebe deferred to future papers.14

Given the difficulty of making accurate CMB polarizationmeasurements at low multipoles, it is useful to investigate other

14See Planck Collaboration Int. XLVI (2016), which has been sub-mitted since this paper was written.

ways of constraining τ. Measurements of the temperature powerspectrum provide a highly accurate measurement of the ampli-tude Ase−2τ. However, as shown in PCP13 CMB lensing breaksthe degeneracy between τ and As. The observed Planck TT spec-trum is, of course, lensed, so the degeneracy between τ and Asis partially broken when we fit models to the Planck TT like-lihood. However, the degeneracy breaking is much stronger ifwe combine the Planck TT likelihood with the Planck lensinglikelihood constructed from measurements of the power spec-trum of the lensing potential Cφφ

`. The 2015 Planck TT and lens-

ing likelihoods are statistically more powerful than their 2013counterparts and the corresponding determination of τ is moreprecise. The 2015 Planck lensing likelihood (labelled “lens-ing”) is summarized in Sec. 5.1 and discussed in more detail inPlanck Collaboration XV (2016). The constraints on τ and zre

15

for various data combinations excluding low-multipole polariza-tion data from Planck are summarized in Fig. 7 and comparedwith the baseline Planck TT+lowP parameters. This figure alsoshows the shifts of other parameters of the base ΛCDM cosmol-ogy, illustrating their sensitivity to changes in τ.

The Planck constraints on τ and zre in the base ΛCDM modelfor various data combinations are:

τ = 0.078+0.019−0.019, zre = 9.9+1.8

−1.6, Planck TT+lowP; (17a)

τ = 0.070+0.024−0.024, zre = 9.0+2.5

−2.1, Planck TT+lensing; (17b)

τ = 0.066+0.016−0.016, zre = 8.8+1.7

−1.4, Planck TT+lowP (17c)+lensing;

τ = 0.067+0.016−0.016, zre = 8.9+1.7

−1.4, Planck TT+lensing (17d)+BAO;

τ = 0.066+0.013−0.013, zre = 8.8+1.3

−1.2, Planck TT+lowP (17e)+lensing+BAO.

15We use the same specific definition of zre as in the 2013 papers,where reionization is assumed to be relatively sharp, with a mid-pointparameterized by a redshift zre and width ∆zre = 0.5. Unless otherwisestated we impose a flat prior on the optical depth with τ > 0.01.

18


0.05 0.10 0.15 0.20

τ

0

5

10

15

20

25

30

35

Pro

bab

ility

den

sity

Planck TT

+lensing

+lensing+BAO

Planck TT+lowP

Planck TT+lowP+WP

Planck TT+lowP+BAO

Fig. 8. Marginalized constraints on the reionization opticaldepth in the base ΛCDM model for various data combina-tions. Solid lines do not include low-multipole polarization; inthese cases the optical depth is constrained by Planck lensing.The dashed/dotted lines include LFI polarization (+lowP), orthe combination of LFI and WMAP polarization cleaned using353 GHz as a dust template (+lowP+WP).

The constraint from Planck TT+lensing+BAO on τ is com-pletely independent of low-multipole CMB polarization data andagrees well with the result from Planck polarization (and hascomparable precision). These results all indicate a lower redshiftof reionization than the value zre = 11.1± 1.1 derived in PCP13,based on the WMAP9 polarization likelihood. The low valuesof τ from Planck are also consistent with the lower value of τderived from the WMAP Planck 353-GHz-cleaned polarizationlikelihood, suggesting strongly that the WMAP9 value is biasedslightly high by residual polarized dust emission.

The Planck results of Eqs. (17a)–(17e) provide evidence fora lower optical depth and redshift of reionization than inferredfrom WMAP (Bennett et al. 2013), partially alleviating the dif-ficulties in reionizing the intergalactic medium using starlightfrom high-redshift galaxies. A key goal of the Planck analysisover the next year is to assess whether these results are consis-tent with the HFI polarization data at low multipoles.

Given the consistency between the LFI and WMAP po-larization maps when both are cleaned with the HFI 353-GHz polarization maps, we have also constructed a combinedWMAP+Planck low-multipole polarization likelihood (denoted“lowP+WP”). This likelihood uses 73 % of the sky and is con-structed from a noise-weighted combination of LFI 70-GHz andWMAP Ka, Q, and V maps, as summarized in Sect. 3.1 anddiscussed in more detail in Planck Collaboration XI (2016). Incombination with the Planck high-multipole TT likelihood, thecombined lowP+WP likelihood gives τ = 0.074+0.011

−0.013, consistentwith the individual LFI and WMAP likelihoods to within about0.5σ.

The various Planck and Planck+WMAP constraints on τ aresummarized in Fig. 8. The tightest of these constraints comesfrom the combined lowP+WP likelihood. It is therefore reason-able to ask why we have chosen to use the lowP likelihood as thebaseline in this paper, which gives a higher statistical error on τ.The principal reason is to produce a Planck analysis, utilizing theLFI polarization data, that is independent of WMAP. All of the

constraints shown in Fig. 8 are compatible with each other, andinsofar as other cosmological parameters are sensitive to smallchanges in τ, it would make very little difference to the resultsin this paper had we chosen to use WMAP or Planck+WMAPpolarization data at low multipoles.

4. Comparison of the Planck power spectrum withhigh-resolution experiments

In PCP13 we combined Planck with the small-scale measure-ments of the ground-based, high-resolution Atacama CosmologyTelescope (ACT) and South Pole Telescope (SPT). The primaryrole of using ACT and SPT was to set limits on foreground com-ponents that were poorly constrained by Planck alone and to pro-vide more accurate constraints on the damping tail of the tem-perature power spectrum. In this paper, with the higher signal-to-noise levels of the full mission Planck data, we have taken a dif-ferent approach, using the ACT and SPT data to impose a prioron the thermal and kinetic SZ power spectrum parameters in thePlanck foreground model as described in Sect. 2.3. In this sec-tion, we check the consistency of the temperature power spectrameasured by Planck, ACT, and SPT, and test the effects of in-cluding the ACT and SPT data on the recovered CMB powerspectrum.

We use the latest ACT temperature power spectra pre-sented in Das et al. (2014), with a revised binning de-scribed in Calabrese et al. (2013) and final beam estimates inHasselfield et al. (2013). As in PCP13 we use ACT data in therange 1 000 < ` < 10 000 at 148 GHz, and 1 500 < ` < 10 000for the 148 × 218 and 218-GHz spectra. We use SPT measure-ments in the range 2 000 < ` < 13 000 from the complete2 540 deg2 SPT-SZ survey at 95, 150, and 220 GHz presentedin George et al. (2015).

Each of these experiments uses a foreground model to de-scribe the multi-frequency power spectra. Here we implement acommon foreground model to combine Planck with the high-multipole data, following a similar approach to PCP13 butwith some refinements. Following the 2013 analysis, we solvefor common nuisance parameters describing the tSZ, kSZ, andtSZ×CIB components, extending the templates used for Planckto ` = 13 000 to cover the full ACT and SPT multipole range.As in PCP13, we use five point-source amplitudes to fit for thetotal dusty and radio Poisson power, namely APS,ACT

148 , APS,ACT218 ,

APS,SPT95 , APS,SPT

150 , and APS,SPT220 . We rescale these amplitudes to

cross-frequency spectra using point-source correlation coeffi-cients, improving on the 2013 treatment by using different pa-rameters for the ACT and SPT correlations, rPS,ACT

148×218 and rPS,SPT150×220

(a single rPS150×220 parameter was used in 2013). We vary rPS,SPT

95×150and rPS,SPT

95×220 as in 2013, and include dust amplitudes for ACT,with Gaussian priors as in PCP13.

As described in Sect. 2.3 we use a theoretically motivatedclustered CIB model fitted to Planck+IRAS estimates of theCIB. The model at all frequencies in the range 95–220 GHz isspecified by a single amplitude ACIB

217 . The CIB power is wellconstrained by Planck data at ` < 2000. At multipoles ` >∼ 3000,the 1-halo component of the CIB model steepens and becomesdegenerate with the Poisson power. This causes an underesti-mate of the Poisson levels for ACT and SPT, inconsistent withpredictions from source counts. We therefore use the PlanckCIB template only in the range 2 < ` < 3000, and extrapo-late to higher multipoles using a power law D` ∝ ` 0.8. Whilethis may not be a completely accurate model for the clustered

19


1000 1500 2000 2500 3000 3500 4000

Multipole moment, `

−100

−50

0

50

100

Res

idua

ls[µ

K2]

Planck

ACT-S

ACT-E

SPT

0.18 0.1 0.072 0.05Angular scale

Fig. 9. Residual power with respect to the Planck TT+lowPΛCDM best-fit model for the Planck (grey), ACT south (or-ange), ACT equatorial (red), and SPT (green) CMB bandpowers.The ACT and SPT bandpowers are scaled by the best-fit calibra-tion factors.

CIB spectrum at high multipoles (see, e.g., Viero et al. 2013;Planck Collaboration XXX 2014), this extrapolation is consis-tent with the CIB model used in the analysis of ACT and SPT.We then need to extrapolate the Planck 217-GHz CIB powerto the ACT and SPT frequencies. This requires converting theCIB measurement in the HFI 217-GHz channel to the ACTand SPT bandpasses assuming a spectral energy distribution; weuse the CIB spectral energy distribution from Bethermin et al.(2012). Combining this model with the ACT and SPT band-passes, we find that ACIB

217 has to be multiplied by 0.12 and 0.89for ACT 148 and 218 GHz, and by 0.026, 0.14, and 0.91 forSPT 95, 150, and 220 GHz, respectively. With this model inplace, the best-fit Planck, ACT, and SPT Poisson levels agreewith those predicted from source counts, as discussed further inPlanck Collaboration XI (2016).

The nuisance model includes seven calibration parametersas in PCP13 (four for ACT and three for SPT). The ACT spec-tra are internally calibrated using the WMAP 9-year maps, with2 % and 7 % uncertainty at 148 and 218 GHz, while SPT cali-brates using the Planck 2013 143-GHz maps, with 1.1 %, 1.2 %,and 2.2 % uncertainty at 95, 150, and 220 GHz. To account forthe increased 2015 Planck absolute calibration (2 % higher inpower) we increase the mean of the SPT map-based calibrationsfrom 1.00 to 1.01.

This common foreground and calibration model fits the datawell. We first fix the cosmology to that of the best-fit PlanckTT+lowP base-ΛCDM model, and estimate the foreground andcalibration parameters, finding a best-fitting χ2 of 734 for 731degrees of freedom (reduced χ2 = 1.004, PTE = 0.46). We thensimultaneously estimate the Planck, ACT- (S: south, E: equa-torial) and SPT CMB bandpowers, Cb, following the Gibbssampling scheme of Dunkley et al. (2013) and Calabrese et al.(2013), marginalizing over the nuisance parameters.

To simultaneously solve for the Planck, ACT, and SPT CMBspectra, we extend the nuisance model described above, includ-

0

500

1000

1500

2000

`2D

TT

`[m

K2]

Planck

Planck+highL

1 0.2 0.1 0.07Angular scale

500 1000 1500 2000 2500

Multipole moment, `

−200

−100

0

100

200

∆D

TT

`[µ

K2]

Fig. 10. Planck CMB power spectrum that is marginalized overforegrounds (red), including a prior on the thermal and kineticSZ power. The inclusion of the full higher resolution ACT andSPT data (shown in blue) does not significantly decrease the er-rors.

ing the four Planck point source amplitudes, the dust parametersand the Planck 100-GHz and 217-GHz calibration parameters(relative to 143 GHz) with the same priors as used in the Planckmulti-frequency likelihood analysis. For ACT and SPT, the cal-ibration factors are defined for each frequency (rather than rel-ative to a central frequency). Following Calabrese et al. (2013),we separate out the 148-GHz calibration for the ACT-(S,E) spec-tra and the 150-GHz calibration for SPT, estimating the CMBbandpowers as Cb/Acal.16 We impose Gaussian priors on Acal:1.00 ± 0.02 for ACT-(S,E); and 1.010 ± 0.012 for SPT. The es-timated CMB spectrum will then have an overall calibration un-certainty for each of the ACT-S, ACT-E, and SPT spectra. We donot require the Planck CMB bandpowers to be the same as thosefor ACT or SPT, so that we can check for consistency betweenthe three experiments.

In Fig. 9 we show the residual CMB power with respect tothe Planck TT+lowP ΛCDM best-fit model for the three experi-ments. All of the data sets are consistent over the multipole rangeplotted in this figure. For ACT-S, we find χ2 = 17.54 (18 datapoints, PTE = 0.49); For ACT-E we find χ2 = 23.54 (18 datapoints, PTE = 0.17); and for SPT χ2 = 5.13 (6 data points,PTE = 0.53).

Figure 10 shows the effect of including ACT and SPT dataon the recovered Planck CMB spectrum. We find that includ-ing the ACT and SPT data does not reduce the Planck errorssignificantly. This is expected because the dominant small-scaleforeground contributions for Planck are the Poisson source am-plitudes, which are treated independently of the Poisson ampli-tudes for ACT and SPT. The high-resolution experiments do help

16This means that the other calibration factors (e.g., ACT 218 GHz)are re-defined to be relative to 148 GHz (or 150 GHz for SPT) data.

20


tighten the CIB amplitude (which is reasonably well constrainedby Planck) and the tSZ and kSZ amplitudes (which are sub-dominant foregrounds for Planck). The kSZ effect in particularis degenerate with the CMB, since both have blackbody compo-nents; imposing a prior on the allowed kSZ power (as discussedin Sect. 2.3) breaks this degeneracy. The net effect is that the er-rors on the recovered Planck CMB spectrum are only marginallyreduced with the inclusion of the ACT and SPT data. This moti-vates our choice to include the information from ACT and SPTinto the joint tSZ and kSZ prior applied to Planck.

The Gibbs sampling technique recovers a best-fit CMB spec-trum marginalized over foregrounds and other nuisance pa-rameters. The Gibbs samples can then be used to form a fastCMB-only Planck likelihood that depends on only one nui-sance parameter, the overall calibration yp. MCMC chains runusing the CMB-only likelihood therefore converge much fasterthan using the full multi-frequency Plik likelihood. The CMB-only likelihood is also extremely accurate, even for extensionsto the base ΛCDM cosmology and is discussed further inPlanck Collaboration XI (2016).

5. Comparison of the Planck base ΛCDM modelwith other astrophysical data sets

5.1. CMB lensing measured by Planck

Gravitational lensing by large-scale structure leaves imprints onthe CMB temperature and polarization that can be measured inhigh angular resolution, low-noise observations, such as thosefrom Planck. The most relevant effects are a smoothing of theacoustic peaks and troughs in the TT , T E, and EE power spec-tra, the conversion of E-mode polarization to B-modes, and thegeneration of significant non-Gaussianity in the form of a non-zero connected 4-point function (see Lewis & Challinor 2006for a review). The latter is proportional to the power spectrumCφφ`

of the lensing potential φ, and so one can estimate thispower spectrum from the CMB 4-point functions. In the 2013Planck release, we reported a 10σ detection of the lensing ef-fect in the TT power spectrum (see PCP13) and a 25σ mea-surement of the amplitude of Cφφ

`from the TTTT 4-point func-

tion (Planck Collaboration XVII 2014). The power of such lens-ing measurements is that they provide sensitivity to parametersthat affect the late-time expansion, geometry, and matter cluster-ing (e.g., spatial curvature and neutrino masses) from the CMBalone.

Since the 2013 Planck release, there have been significantdevelopments in the field of CMB lensing. The SPT team havereported a 7.7σ detection of lens-induced B-mode polariza-tion based on the EBφCIB 3-point function, where φCIB is aproxy for the CMB lensing potential φ derived from CIB mea-surements (Hanson et al. 2013). The POLARBEAR collabora-tion (POLARBEAR Collaboration 2014b) and the ACT collabo-ration (van Engelen et al. 2015) have performed similar analysesat somewhat lower significance (POLARBEAR Collaboration2014b). In addition, the first detections of the polarization 4-point function from lensing, at a significance of around 4σ,have been reported by the POLARBEAR (Ade et al. 2014) andSPT (Story et al. 2015) collaborations, and the former have alsomade a direct measurement of the BB power spectrum dueto lensing on small angular scales with a significance around2σ (POLARBEAR Collaboration 2014a). Finally, the BB powerspectrum from lensing has also been detected on degree angu-lar scales, with similar significance, by the BICEP2 collabora-tion (BICEP2 Collaboration 2014); see also BKP.

5.1.1. The Planck lensing likelihood

Lensing results from the full-mission Planck data are discussedin Planck Collaboration XV (2016).17 With approximately twicethe amount of temperature data, and the inclusion of polariza-tion, the noise levels on the reconstructed φ are a factor of about2 better than in Planck Collaboration XVII (2014). The broad-band amplitude of Cφφ

ìs now measured to better than 2.5 %

accuracy, the most significant measurement of CMB lensingto date. Moreover, lensing B-modes are detected at 10σ, boththrough a correlation analysis with the CIB and via the TT EB4-point function. Many of the results in this paper make use ofthe Planck measurements of Cφφ

`. In particular, they provide an

alternative route to estimate the optical depth (as already dis-cussed in Sect. 3.4), and to tightly constrain spatial curvature(Sect. 6.2.4).

The estimation of Cφφ`

from the Planck full-mission data isdiscussed in detail in Planck Collaboration XV (2016). There area number of significant changes from the 2013 analysis that areworth noting here.

• The lensing potential power spectrum is now estimated fromlens reconstructions that use both temperature and polariza-tion data in the multipole range 100 ≤ ` ≤ 2048. The like-lihood used here is based on the power spectrum of a lensreconstruction derived from the minimum-variance combi-nation of five quadratic estimators (TT , T E, EE, T B, andEB). The power spectrum is therefore based on 15 different4-point functions.

• The results used here are derived from foreground-cleanedmaps of the CMB synthesized from all nine Planck fre-quency maps with the SMICA algorithm, while the baseline2013 results used a minimum-variance combination of the143-GHz and 217-GHz nominal-mission maps. After mask-ing the Galaxy and point-sources, 67.3 % of the sky is re-tained for the lensing analysis.

• The lensing power spectrum is estimated in the multipolerange 8 ≤ ` ≤ 2048. Multipoles ` < 8 have large mean-fieldcorrections due to survey anisotropy and are rather unsta-ble to analysis choices; they are therefore excluded from alllensing results. Here, we use only the range 40 ≤ ` ≤ 400(the same as used in the 2013 analysis), with eight binseach of width ∆` = 45. This choice is based on the exten-sive suite of null tests reported in Planck Collaboration XV(2016). Nearly all tests are passed over the full multipolerange 8 ≤ ` ≤ 2048, with the exception of a slight excess ofcurl modes in the TT reconstruction around ` = 500. Giventhat the range 40 ≤ ` ≤ 400 contains most of the statisticalpower in the reconstruction, we have conservatively adoptedthis range for use in the Planck 2015 cosmology papers.

• To normalize Cφφ`

from the measured 4-point functions re-quires knowledge of the CMB power spectra. In practice, wenormalize with fiducial spectra, but then correct for changesin the true normalization at each point in parameter spacewithin the likelihood. The exact renormalization schemeadopted in the 2013 analysis proved to be too slow for theextension to polarization, so we now use a linearized approx-imation, based on pre-computed response functions, which isvery efficient within an MCMC analysis. Spot-checks haveconfirmed the accuracy of this approach.

17In that paper we are careful to highlight the 4-point function ori-gin of the lensing power spectrum reconstruction by using the index L;however, in this paper we use the notation `.

21


50 100 150 200 250 300 350 400

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[`(`

+1)

]2Cφφ

`/2π

[×10

7]

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T + P

50 100 150 200 250 300 350 400

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−1

01

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+1)

]2∆

Cφφ

`/2π

[×10

8]

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+1)

]2Cφφ

`/

2π[×

107]

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−1

01

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[`(`

+1)

]2∆

Cφφ

`/

2π[×

108]

Fig. 11. Planck measurements of the lensing power spectrum compared to the prediction for the best-fitting base ΛCDM model tothe Planck TT+lowP data. Left: the conservative cut of the Planck lensing data used throughout this paper, covering the multipolerange 40 ≤ ` ≤ 400. Right: lensing data over the range 8 ≤ ` ≤ 2048, demonstrating the general consistency with the ΛCDMprediction over this extended multipole range. In both cases, green points are the power from lensing reconstructions using onlytemperature data, while blue points combine temperature and polarization. They are offset in ` for clarity and error bars are ±1σ. Inthe top panels the solid lines are the best-fitting base ΛCDM model to the Planck TT+lowP data with no renormalization or δN(1)

correction applied (see text for explanation). The bottom panels show the difference between the data and the renormalized andδN(1)-corrected theory bandpowers, which enter the likelihood. The mild preference of the lensing measurements for lower lensingpower around ` = 200 pulls the theoretical prediction for Cφφ

`downwards at the best-fitting parameters of a fit to the combined

Planck TT+lowP+lensing data, shown by the dashed blue lines (always for the conservative cut of the lensing data, includingpolarization).

• The measurement of Cφφ`

can be thought of as being de-rived from an optimal combination of trispectrum configu-rations. In practice, the expectation value of this combina-tion at any multipole ` has a local part proportional to Cφφ

`,

but also a non-local (“N(1) bias”) part that couples to a broadrange of multipoles in Cφφ

`(Kesden et al. 2003); this non-

local part comes from non-primary trispectrum couplings. Inthe Planck 2013 analysis we corrected for the N(1) bias bymaking a fiducial correction, but this ignores its parameterdependence. We improve on this in the 2015 analysis by cor-recting for errors in the fiducial N(1) bias at each point inparameter space within the lensing likelihood. As with therenormalization above, we linearize this δN(1) correction forefficiency. As a result, we no longer need to make an approx-imate correction in the Cφφ

`covariance matrix to account for

the cosmological uncertainty in N(1).• Beam uncertainties are no longer included in the covariance

matrix of Cφφ`

, since, with the improved knowledge of thebeams, the estimated uncertainties are negligible for the lens-ing analysis. The only inter-bandpower correlations includedin the Cφφ

`bandpower covariance matrix are from the uncer-

tainty in the correction applied for the point-source 4-pointfunction.

As in the 2013 analysis, we approximate the lensing likelihoodas Gaussian in the estimated bandpowers, with a fiducial co-variance matrix. Following the arguments in Schmittfull et al.(2013), it is a good approximation to ignore correlations betweenthe 2- and 4-point functions; so, when combining the Planckpower spectra with Planck lensing, we simply multiply their re-spective likelihoods.

It is also worth noting that the changes in absolute calibra-tion of the Planck power spectra (around 2 % between the 2013and 2015 releases) do not directly affect the lensing results. TheCMB 4-point functions do, of course, respond to any recalibra-tion of the data, but in estimating Cφφ

`this dependence is re-

moved by normalizing with theory spectra fit to the observedCMB spectra. The measured Cφφ

`bandpowers from the 2013 and

current Planck releases can therefore be directly compared, andare in good agreement (Planck Collaboration XV 2016). Care isneeded, however, in comparing consistency of the lensing mea-surements across data releases with the best-fitting model pre-dictions. Changes in calibration translate directly into changesin Ase−2τ, which, along with any change in the best-fitting opti-cal depth, alter As, and hence the predicted lensing power. Thesechanges from 2013 to the current release go in opposite direc-tions, leading to a net decrease in As of 0.6 %. This, combinedwith a small (0.15 %) increase in θeq, reduces the expected Cφφ

`by approximately 1.5 % for multipoles ` > 60.

The Planck measurements of Cφφ`

, based on the temperatureand polarization 4-point functions, are plotted in Fig. 11 (withresults of a temperature-only reconstruction included for com-parison). The measured Cφφ

àre compared with the predicted

lensing power from the best-fitting base ΛCDM model to thePlanck TT+lowP data in this figure. The bandpowers that areused in the conservative lensing likelihood adopted in this pa-per are shown in the left-hand plot, while bandpowers over therange 8 ≤ ` ≤ 2048 are shown in the right-hand plot, to demon-strate the general consistency with the ΛCDM prediction overthe full multipole range. The difference between the measuredbandpowers and the best-fit prediction are shown in the bottom

22


panels. Here, the theory predictions are corrected in the sameway as they are in the likelihood.18

Figure 11 suggests that the Planck measurements of Cφφ`

aremildly in tension with the prediction of the best-fitting ΛCDMmodel. In particular, for the conservative multipole range 40 ≤` ≤ 400, the temperature+polarization reconstruction has χ2 =15.4 (for eight degrees of freedom), with a PTE of 5.2 %. Forreference, over the full multipole range χ2 = 40.8 for 19 degreesof freedom (PTE of 0.3 %); the large χ2 is driven by a singlebandpower (638 ≤ ` ≤ 762), and excluding this gives an ac-ceptable χ2 = 26.8 (PTE of 8 %). We caution the reader thatthis multipole range is where the lensing reconstruction shows amild excess of curl-modes (Planck Collaboration XV 2016), andfor this reason we adopt the conservative multipole range for thelensing likelihood in this paper.

This simple χ2 test does not account for the uncertainty inthe predicted Cφφ

`. In the ΛCDM model, the dominant uncer-

tainty in the multipole range 40 ≤ ` ≤ 400 comes from that inAs (1σ uncertainty of 3.7 % for Planck TT+lowP), which itselfderives from the uncertainty in the reionization optical depth, τ.The predicted rms lensing deflection from Planck TT+lowP datais 〈d2〉1/2 = (2.50 ± 0.05) arcmin, corresponding to a 3.6 % un-certainty (1σ) in the amplitude of Cφφ

`(which improves to 3.1 %

uncertainty for the combined Planck+WP likelihood). Note thatthis is larger than the uncertainty on the measured amplitude,i.e., the lensing measurement is more precise than the predictionfrom the CMB power spectra in even the simplest ΛCDM model.This model uncertainty is reflected in a scatter in the χ2 of thelensing data over the Planck TT+lowP chains, χ2

lens = 17.9±9.0,which is significantly larger than the expected scatter in χ2 at thetrue model, due to the uncertainties in the lensing bandpowers(√

2Ndof = 4). Following the treatment in PCP13, we can assessconsistency more carefully by introducing a parameter Aφφ

L thatscales the theory lensing trispectrum at every point in parame-ter space in a joint analysis of the CMB spectra and the lensingspectrum. We find

AφφL = 0.95 ± 0.04 (68%,Planck TT+lowP+lensing), (18)

in good agreement with the expected value of unity. The pos-terior for Aφφ

L , and other lensing amplitude measures discussedbelow, is shown in Fig. 12.

Given the precision of the measured Cφφ`

compared to theuncertainty in the predicted spectrum from fits to the PlanckTT+lowP data, the structure in the residuals seen in Fig. 11might be expected to pull parameters in joint fits. As discussedin Planck Collaboration XV (2016) and Pan et al. (2014), theprimary parameter dependence of Cφφ

àt multipoles ` >∼ 100

is through As and èq in ΛCDM models. Here, èq ∝ 1/θeq is theangular multipole corresponding to the horizon size at matter-radiation equality observed at a distance χ∗. The combinationAsèq determines the mean-squared deflection 〈d2〉, while èq

controls the shape of Cφφ`

. For the parameter ranges of interest,

δCφφ`/Cφφ

`= δAs/As + (n` + 1)δèq/èq, (19)

18In detail, the theory spectrum is binned in the same way as thedata, renormalized to account for the (very small) difference betweenthe CMB spectra in the best-fit model and the fiducial spectra used in thelensing analysis, and corrected for the difference in N(1), calculated forthe best-fit and fiducial models (around a 4 % change in N(1), since thefiducial-model Cφφ

` is higher by this amount than in the best-fit model).

0.6 1.0 1.4 1.8 2.2

AL

0

2

4

6

8

10

Pro

bab

ility

den

sity

Planck TT+lowP

+lensing (AφφL )

Planck TE+lowP

Planck EE+lowP


Fig. 12. Marginalized posterior distributions for measures of thelensing power amplitude. The dark-blue (dot-dashed) line is theconstraint on the parameter Aφφ

L , which scales the amplitude ofthe lensing power spectrum in the lensing likelihood for thePlanck TT+lowP+lensing data combination. The other lines arefor the AL parameter, which scales the lensing power spectrumused to lens the CMB spectra, for the data combinations PlanckTT+lowP (blue, solid), Planck TE+lowP (red, dashed), PlanckEE+lowP (green, dashed), and Planck TT,TE,EE+lowP (black,dashed). The dotted lines show the AL constraints when the Pliklikelihood is replaced with CamSpec, highlighting that the pref-erence for high AL in the Planck EE+lowP data combination isnot robust to the treatment of polarization on intermediate andsmall scales.

where n` arises (mostly) from the strong wavenumber depen-dence of the transfer function for the gravitational potential, withn` ≈ 1.5 around ` = 200.

In joint fits to Planck TT+lowP+lensing, the main param-eter changes from Planck TT+lowP alone are a 2.6 % reduc-tion in the best-fit As, with an accompanying reduction in thebest-fit τ, to 0.067 (around 0.6σ; see Sect. 3.4). There is alsoa 0.7 % reduction in èq, achieved at fixed θ∗ by reducing ωm.These combine to reduce Cφφ

`by approximately 4 % at ` = 200,

consistent with Eq. (19). The difference between the theory lens-ing spectrum at the best-fit parameters in the Planck TT+lowPand Planck TT+lowP+lensing fits are shown by the dashed bluelines in Fig. 11. In the joint fit, the χ2 for the lensing bandpow-ers improves by 6, while the χ2 for the Planck TT+lowP datadegrades by only 1.2 (2.8 for the high-` TT data and −1.6 forthe low-` T EB data).

The lower values of As and ωm in the joint fit give a 2 %reduction in σ8, with

σ8 = 0.815 ± 0.009 (68%,Planck TT+lowP+lensing), (20)

as shown in Fig. 19. The decrease in matter density leads to acorresponding decrease in Ωm, and at fixed θ∗ (approximately∝ Ωmh3) a 0.5σ increase in H0, giving

H0 = (67.8 ± 0.9) km s−1Mpc−1

Ωm = 0.308 ± 0.012

Planck TT+lowP+lensing.

(21)Joint Planck+lensing constraints on other parameters of the baseΛCDM cosmology are given in Table. 4.

23


Planck Collaboration XV (2016) discusses the effect on pa-rameters of extending the lensing multipole range in joint fitswith Planck TT+lowP. In the base ΛCDM model, using thefull multipole range 8 ≤ ` ≤ 2048, the parameter combinationσ8Ω

1/4m ≈ (As`

2.5eq )1/2 (which is well determined by the lensing

measurements) is pulled around 1σ lower that its value usingthe conservative lensing range, with a negligible change in theuncertainty. Around half of this shift comes from the 3.6σ out-lying bandpower (638 ≤ ` ≤ 762). In massive neutrino models,the total mass is similarly pulled higher by around 1σ when us-ing the full lensing multipole range.

5.1.2. Detection of lensing in the CMB power spectra

The smoothing effect of lensing on the acoustic peaksand troughs of the TT power spectrum is detected athigh significance in the Planck data. Following PCP13 (seealso Calabrese et al. 2008), we introduce a parameter AL, whichscales the theory Cφφ

`power spectrum at each point in param-

eter space, and which is used to lens the CMB spectra.19 Theexpected value for base ΛCDM is AL = 1. The results of suchan analysis for models with variable AL is shown in Fig. 12. Themarginalized constraint on AL is

AL = 1.22 ± 0.10 (68%,Planck TT+lowP) . (22)

This is very similar to the result from the 2013 Planck data re-ported in PCP13. The persistent preference for AL > 1 is dis-cussed in detail there. For the 2015 data, we find that ∆χ2 = −6.4between the best-fitting ΛCDM+AL model and the best-fittingbase ΛCDM model. There is roughly equal preference for highAL from intermediate and high multipoles (i.e., the Plik likeli-hood; ∆χ2 = −2.6) and from the low-` likelihood (∆χ2 = −3.1),with a further small change coming from the priors.

Increases in AL are accompanied by changes in all other pa-rameters, with the general effect being to reduce the predictedCMB power on large scales, and in the region of the secondacoustic peak, and to increase CMB power on small scales (seeFig. 13). A reduction in the high-` foreground power compen-sates the CMB increase on small scales. Specifically, ns is in-creased by 1 % relative to the best-fitting base model and As isreduced by 4 %, both of which lower the large-scale power toprovide a better fit to the measured spectra around ` = 20 (seeFig. 1). The densities ωb and ωc respond to the change in ns, fol-lowing the usual ΛCDM acoustic degeneracy, and Ase−2τ fallsby 1 %, attempting to reduce power in the damping tail due tothe increase in ns and reduction in the diffusion angle θD (whichfollows from the reduction in ωm). The changes in As and Ase−2τ

lead to a reduction in τ from 0.078 to 0.060. With these cos-mological parameters, the lensing power is lower than in thebase model, which additionally increases the CMB power in theacoustic peaks and reduces it in the troughs. This provides a poorfit to the measured spectra around the fourth and fifth peaks, butthis can be mitigated by increasing AL to give more smoothingfrom lensing than in the base model. However, AL further in-creases power in the damping tail, but this is partly offset byreduction of the power in the high-` foregrounds.

19We emphasize the difference between the phenomenological pa-rameters AL and Aφφ

L (introduced earlier). The amplitude AL multipliesCφφ` when calculating both the lensed CMB theory spectra and the lens-

ing likelihood, while AφφL affects only the lensing likelihood by scaling

the theory Cφφ` when comparing with the power spectrum of the recon-

structed lensing potential φ.

500 1000 1500 2000 2500Multipole `

−50

−25

025

50

∆D

TT

`[µ

K2]

100×100 foregrounds

143×143

143×217

217×217

Fig. 13. Changes in the CMB TT spectrum and foreground spec-tra, between the best-fitting AL model and the best-fitting baseΛCDM model to the Planck TT+lowP data. The solid blue lineshows the difference between the AL model and ΛCDM whilethe dashed line has the the same values of the other cosmologi-cal parameters, but with AL set to unity, to highlight the changesin the spectrum arising from differences in the other parame-ters. Also shown are the changes in the best-fitting foregroundcontributions to the four frequency cross-spectra between the ALmodel and the ΛCDM model. The data points (with ±1σ er-rors) are the differences between the high-`maximum-likelihoodfrequency-averaged CMB spectrum and the best-fitting ΛCDMmodel to the Planck TT+lowP data (as in Fig. 1). Note that thechanges in the CMB spectrum and the foregrounds should beadded when comparing to the residuals in the data points.

The trends in the TT spectrum that favour high AL have asimilar pull on parameters such as curvature (Sect. 6.2.4) andthe dark energy equation of state (Sect. 6.3) in extended models.These parameters affect the late-time geometry and clusteringand so alter the lensing power, but their effect on the primaryCMB fluctuations is degenerate with changes in the Hubble con-stant (to preserve θ∗). The same parameter changes as those inAL models are found in these extended models, but with, for ex-ample, the increase in AL replaced by a reduction in ΩK . Addingexternal data, however, such as the Planck lensing data or BAO(Sect. 5.2), pull these extended models back to base ΛCDM.

Finally, we note that lensing is also detected at lower signif-icance in the polarization power spectra (see Fig. 12):

AL = 0.98+0.21−0.24 (68%,Planck TE+lowP) ; (23a)

AL = 1.54+0.28−0.33 (68%,Planck EE+lowP) . (23b)

These results use only polarization at low multipoles, i.e., withno temperature data at multipoles ` < 30. These are the first de-tections of lensing in the CMB polarization spectra, and reachalmost 5σ in T E. We caution the reader that the AL constraintsfrom EE and low-` polarization are rather unstable betweenhigh-` likelihoods because of differences in the treatment of thepolarization data (see Fig. 12, which compares constraints fromthe Plik and CamSpec polarization likelihoods). The result ofreplacing Plik with the CamSpec likelihood is AL = 1.19+0.20

−0.24,i.e., around 1σ lower than the result from Plik reported inEq. (23b). If we additionally include the low-` temperature data,AL from T E increases:

AL = 1.13 ± 0.2 (68%,Planck TE+lowT,P) . (24)

24


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

z

0.90

0.95

1.00

1.05

1.10

(DV/

r dra

g)/

(DV/

r dra

g) P

lan

ck

6DFGS

SDSS MGS

BOSS LOWZBOSS CMASS

WiggleZ

Fig. 14. Acoustic-scale distance ratio DV(z)/rdrag in the baseΛCDM model divided by the mean distance ratio from PlanckTT+lowP+lensing. The points with 1σ errors are as follows:green star (6dFGS, Beutler et al. 2011); square (SDSS MGS,Ross et al. 2015); red triangle and large circle (BOSS “LOWZ”and CMASS surveys, Anderson et al. 2014); and small blue cir-cles (WiggleZ, as analysed by Kazin et al. 2014). The grey bandsshow the 68 % and 95 % confidence ranges allowed by PlanckTT+lowP+lensing.

The pull to higher AL in this case is due to the reduction in TTpower in these models on large scales (as discussed above).

5.2. Baryon acoustic oscillations

Baryon acoustic oscillation (BAO) measurements are geometricand largely unaffected by uncertainties in the nonlinear evolutionof the matter density field and additional systematic errors thatmay affect other types of astrophysical data. As in PCP13, wetherefore use BAO as a primary astrophysical data set to breakparameter degeneracies from CMB measurements.

Figure 14 shows an updated version of figure 15 fromPCP13. The plot shows the acoustic-scale distance ratioDV(z)/rdrag measured from a number of large-scale struc-ture surveys with effective redshift z, divided by the meanacoustic-scale ratio in the base ΛCDM cosmology using PlanckTT+lowP+lensing. Here rdrag is the comoving sound horizon atthe end of the baryon drag epoch and DV is a combination of theangular diameter distance DA(z) and Hubble parameter H(z),

DV(z) =

[(1 + z)2D2

A(z)cz

H(z)

]1/3

. (25)

The grey bands in the figure show the ±1σ and ±2σ rangesallowed by Planck in the base ΛCDM cosmology.

The changes to the data points compared to figure 15 ofPCP13 are as follows. We have replaced the SDSS DR7 mea-surements of Percival et al. (2010) with the recent analysis ofthe SDSS Main Galaxy Sample (MGS) of Ross et al. (2015)at zeff = 0.15, and by the Anderson et al. (2014) analysis ofthe Baryon Oscillation Spectroscopic Survey (BOSS) “LOWZ”sample at zeff = 0.32. Both of these analyses use peculiar veloc-ity field reconstructions to sharpen the BAO feature and reduce

1350 1400 1450 1500

DA(0.57)(r fiddrag/rdrag) [Mpc]

85

90

95

100

105

110

H(0

.57

)(r d

rag/

rfid

dra

g)

[km

s−1M

pc−

1] BOSS CMASS

0.1125

0.1140

0.1155

0.1170

0.1185

0.1200

0.1215

0.1230

0.1245

Ωc h

2

Fig. 15. 68 % and 95 % constraints on the angular diameter dis-tance DA(z = 0.57) and Hubble parameter H(z = 0.57) fromthe Anderson et al. (2014) analysis of the BOSS CMASS-DR11sample. The fiducial sound horizon adopted by Anderson et al.(2014) is rfid

drag = 149.28 Mpc. Samples from the PlanckTT+lowP+lensing chains are plotted coloured by their value ofΩch2, showing consistency of the data, but also that the BAOmeasurement can tighten the Planck constraints on the matterdensity.

the errors on DV/rdrag. The blue points in Fig. 14 show a reanal-ysis of the WiggleZ redshift survey by Kazin et al. (2014) thatapplyies peculiar velocity reconstructions. These reconstructionscause small shifts in DV/rdrag compared to the unreconstructedWiggleZ results of Blake et al. (2011) and lead to reductionsin the errors on the distance measurements at zeff = 0.44 andzeff = 0.73. The point labelled BOSS CMASS at zeff = 0.57shows DV/rdrag from the analysis of Anderson et al. (2014), up-dating the BOSS-DR9 analysis of Anderson et al. (2012) used inPCP13.

In fact, the Anderson et al. (2014) analysis solves jointly forthe positions of the BAO feature in both the line-of-sight andtransverse directions (the distortion in the transverse directioncaused by the background cosmology is sometimes called theAlcock-Paczynski effect, Alcock & Paczynski 1979), leading tojoint constraints on the angular diameter distance DA(zeff) andthe Hubble parameter H(zeff). These constraints, using the tabu-lated likelihood included in the CosmoMC module,20 are plottedin Fig. 15. Samples from the Planck TT+lowP+lensing chainsare shown for comparison, coloured by the value of Ωch2. Thelength of the degeneracy line is set by the allowed variation in H0(or equivalently Ωmh2). In the Planck TT+lowP+lensing ΛCDManalysis the line is defined approximately by

DA(0.57)/rdrag

9.384

(H(0.57)rdrag/c

0.4582

)1.7

= 1.0000 ± 0.0004, (26)

which just grazes the BOSS CMASS 68 % error ellipse plottedin Fig. 15. Evidently, the Planck base ΛCDM parameters arein good agreement with both the isotropized DV BAO measure-ments plotted in Fig. 14, and with the anisotropic constraintsplotted in Fig. 15.

20http://www.sdss3.org/science/boss_publications.php

25

http://www.sdss3.org/science/boss_publications.php


In this paper, we use the 6dFGS, SDSS-MGS, and BOSS-LOWZ BAO measurements of DV/rdrag (Beutler et al. 2011;Ross et al. 2015; Anderson et al. 2014) and the CMASS-DR11anisotropic BAO measurements of Anderson et al. (2014). Sincethe WiggleZ volume partially overlaps that of the BOSS-CMASS sample, and the correlations have not been quantified,we do not use the WiggleZ results in this paper. It is clear fromFig. 14 that the combined BAO likelihood is dominated by thetwo BOSS measurements.

In the base ΛCDM model, the Planck data constrain theHubble constant H0 and matter density Ωm to high precision:

H0 = (67.3 ± 1.0) km s−1Mpc−1

Ωm = 0.315 ± 0.013

Planck TT+lowP. (27)

With the addition of the BAO measurements, these constraintsare strengthened significantly to

H0 = (67.6 ± 0.6) km s−1Mpc−1

Ωm = 0.310 ± 0.008

Planck TT+lowP+BAO.

(28)These numbers are consistent with the Planck+lensing con-straints of Eq. (21). Section 5.4 discusses the consistency ofthese estimates of H0 with direct measurements.

Although low-redshift BAO measurements are in goodagreement with Planck for the base ΛCDM cosmology, thismay not be true at high redshifts. Recently, BAO features havebeen measured in the flux-correlation function of the Lyα for-est of BOSS quasars (Delubac et al. 2015) and in the cross-correlation of the Lyα forest with quasars (Font-Ribera et al.2014). These observations give measurements of c/(H(z)rdrag)and DA(z)/rdrag (with somewhat lower precision) at z = 2.34and z = 2.36, respectively. For example, from table II ofAubourg et al. (2015) the two Lyα BAO measurements com-bined give c/(H(2.34)rdrag) = 9.14 ± 0.20, compared to the pre-dictions of the base Planck ΛCDM cosmology of 8.586± 0.021,which are discrepant at the 2.7σ level. At present, it is not clearwhether this discrepancy is caused by systematics in the LyαBAO measurements (which are more complex and less maturethan galaxy BAO measurements) or an indicator of new physics.As Aubourg et al. (2015) discuss, it is difficult to find a physicalexplanation for the Lyα BAO results without disrupting the con-sistency with the much more precise galaxy BAO measurementsat lower redshifts.

5.3. Type Ia supernovae

Type Ia supernovae (SNe) are powerful probes of cosmology(Riess et al. 1998; Perlmutter et al. 1999) and particularly of theequation of state of dark energy. In PCP13, we used two sam-ples of type Ia SNe, the “SNLS” compilation (Conley et al.2011) and the “Union2.1” compilation (Suzuki et al. 2012). TheSNLS sample was found to be in mild tension, at about the 2σlevel, with the 2013 Planck base ΛCDM cosmology favouringa value of Ωm ≈ 0.23 compared to the Planck value of Ωm =0.315±0.017. Another consequence of this tension showed up inextensions to the base ΛCDM model, where the combination ofPlanck and the SNLS sample showed 2σ evidence for a “phan-tom” (w < −1) dark energy equation of state.

Following the submission of PCP13, Betoule et al. (2013) re-ported the results of an extensive campaign to improve the rel-ative photometric calibrations between the SNLS and SDSS su-pernova surveys. The “Joint Light-curve Analysis” (JLA) sam-ple, used in this paper, is constructed from the SNLS and SDSS

SNe data, together with several samples of low redshift SNe.21.Cosmological constraints from the JLA sample are discussedby Betoule et al. (2014) and residual biases associated with thephotometry and light curve fitting are assessed by Mosher et al.(2014). For the base ΛCDM cosmology, Betoule et al. (2014)find Ωm = 0.295 ± 0.034, consistent with the 2013 and 2015Planck values for base ΛCDM. This relieves the tension betweenthe SNLS and Planck data reported in PCP13. Given the consis-tency between Planck and the JLA sample for base ΛCDM, onecan anticipate that the combination of these two data sets willconstrain the dark energy equation of state to be close to w = −1(see Sect. 6.3).

Since the submission of PCP13, first results from a sampleof Type Ia SNe discovered with the Pan-STARRS survey havebeen reported by Rest et al. (2014) and Scolnic et al. (2014). ThePan-STARRS sample is still relatively small (consisting of 146spectroscopically confirmed Type Ia SNe) and is not used in thispaper.

5.4. The Hubble constant

CMB experiments provide indirect and highly model-dependentestimates of the Hubble constant. It is therefore important tocompare CMB estimates with direct estimates of H0, since anysignificant evidence of a tension could indicate the need for newphysics. In PCP13, we used the Riess et al. (2011, hereafter R11)Hubble Space Telescope (HST) Cepheid+SNe based estimate ofH0 = (73.8 ± 2.4) km s−1Mpc−1 as a supplementary “H0-prior.”This value was in tension at about the 2.5σ level with the 2013Planck base ΛCDM value of H0.

For the base ΛCDM model, CMB and BAO experimentsconsistently find a value of H0 lower than the R11 value.For example, the 9-year WMAP data (Bennett et al. 2013;Hinshaw et al. 2013) give:22

H0 = (69.7 ± 2.1) km s−1Mpc−1, WMAP9, (29a)H0 = (68.0 ± 0.7) km s−1Mpc−1, WMAP9+BAO. (29b)

These numbers can be compared with the Planck 2015 valuesgiven in Eqs. (27) and (28). The WMAP constraints are driventowards the Planck values by the addition of the BAO data and sothere is persuasive evidence for a low H0 in the base ΛCDM cos-mology independently of the high-multipole CMB results fromPlanck. The 2015 Planck TT+lowP value is entirely consistentwith the 2013 Planck value and so the tension with the R11 H0determination remains at about 2.4σ.

The tight constraint on H0 in Eq. (29b) is an example of an“inverse distance ladder,” where the CMB primarily constrainsthe sound horizon within a given cosmology, providing an ab-solute calibration of the BAO acoustic-scale (e.g., Percival et al.2010; Cuesta et al. 2015; Aubourg et al. 2015, see also PCP13).In fact, in a recent paper Aubourg et al. (2015) use the 2013Planck constraints on rs in combination with BAO and the JLASNe data to find H0 = (67.3 ± 1.1) km s−1Mpc−1, in excellentagreement with the 2015 Planck value for base ΛCDM given in

21A CosmoMC likelihood model for the JLA sample is avail-able at http://supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html. The latest version in CosmoMC includes numerical integrationover the nuisance parameters for use when calculating joint constraintsusing importance sampling; this can give different χ2 values comparedto parameter best fits.

22These numbers are taken from our parameter grid, which includesa neutrino mass of 0.06 eV and the same updated BAO compilation asEq. (28) (see Sect. 5.2).

26

http://supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html

http://supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html


Eq. (27), which is based on the Planck temperature power spec-trum. Note that by adding SNe data, the Aubourg et al. (2015)estimate of H0 is insensitive to spatial curvature and to late timevariations of the dark energy equation of state. Evidently, thereare a number of lines of evidence that point to a lower value ofH0 than the direct determination of R11.

The R11 Cepheid data have been reanalysed by Efstathiou(2014, hereafter E14) using the revised geometric maser distanceto NGC 4258 of Humphreys et al. (2013). Using NGC 4258 as adistance anchor, E14 finds

H0 = (70.6 ± 3.3) km s−1Mpc−1, NGC 4258, (30)

which is within 1σ of the Planck TT estimate given in Eq. (27).In this paper we use Eq. (30) as a “conservative” H0 prior.

R11 also use Large Magellanic Cloud Cepheids and a smallsample of Milky Way Cepheids with parallax distances as al-ternative distance anchors to NGC4258. The R11 H0 prior usedin PCP13 combines all three distance anchors. Combining theLMC and MW distance anchors, E14 finds

H0 = (73.9 ± 2.7) km s−1Mpc−1, LMC + MW, (31)

under the assumption that there is no metallicity variation ofthe Cepheid period-luminosity relation. This is discrepant withEq. (27) at about the 2.2σ level. However, neither the centralvalue nor the error in Eq. (31) is reliable. The MW Cepheidsample is small and dominated by short period (< 10 day) ob-jects. The MW Cepheid sample therefore has very little over-lap with the period range of SNe host galaxy Cepheids observedwith HST. As a result, the MW solutions for H0 are unstable(see Appendix A of E14). The LMC solution is sensitive to themetallicity dependence of the Cepheid period-luminosity rela-tion which is poorly constrained by the R11 data. Furthermore,the estimate in Eq. (30) is based on a differential measurement,comparing HST photometry of Cepheids in NGC 4258 withthose in SNe host galaxies. It is therefore less prone to pho-tometric systematics, such as crowding corrections, than is theLMC+MW estimate of Eq. (31). It is for these reasons that wehave adopted the prior of Eq. (30) in preference to using theLMC and MW distance anchors.23

Direct measurements of the Hubble constant have a long andsometimes contentious history (see, e.g., Tammann et al. 2008).The controversy continues to this day and in the literature onecan find “high” values, e.g., H0 = (74.3 ± 2.6) km s−1Mpc−1

(Freedman et al. 2012), and “low” values, e.g., H0 = (63.7 ±2.3) km s−1Mpc−1 (Tammann & Reindl 2013). The key pointthat we wish to make is that the Planck-only estimates ofEqs. (21) and (27), and the Planck+BAO estimate of Eq. (28)all have small errors and are consistent. If a persuasive case canbe made that a direct measurement of H0 conflicts with these es-timates, then this will be strong evidence for additional physicsbeyond the base ΛCDM model.

23As this paper was nearing completion, results from the NearbySupernova Factory have been presented that indicate a correlation be-tween the peak brightness of Type Ia SNe and the local star-formationrate (Rigault et al. 2015). These authors argue that this correlation intro-duces a systematic bias of around 1.8 km s−1Mpc−1 in the SNe/Cepheiddistance scale measurement of H0 . For example, according to theseauthors, the estimate of Eq. (30) should be lowered to H0 = (68.8 ±3.3) km s−1Mpc−1, a downward shift of approximately 0.5σ. Clearly,further work needs to be done to assess the importance of such a biason the distance scale. It is ignored in the rest of this paper.

Finally, we note that in a recent analysis Bennett et al. (2014)derive a “concordance” value of H0 = (69.6±0.7) km s−1Mpc−1

for base ΛCDM by combining WMAP9+SPT+ACT+BAOwith a slightly revised version of the R11 H0 value, (73.0 ±2.4) km s−1Mpc−1. The Bennett et al. (2014) central value forH0 differs from the Planck value of Eq. (28) by nearly 3 % (or2.5σ). The reason for this difference is that the Planck data arein tension with the Story et al. (2013) SPT data (as discussed inAppendix B of PCP13; note that the tension is increased with thePlanck full mission data) and with the revised R11 H0 determi-nation. Both tensions drive the Bennett et al. (2014) value of H0away from the Planck solution.

5.5. Additional data

5.5.1. Redshift space distortions

Transverse versus line-of-sight anisotropies in the redshift-spaceclustering of galaxies induced by peculiar motions can, poten-tially, provide a powerful way of constraining the growth rate ofstructure (e.g., Percival & White 2009). A number of studies ofredshift-space distortions (RSD) have been conducted to mea-sure the parameter combination fσ8(z), where for models withscale-independent growth

f (z) =d ln Dd ln a

, (32)

and D is the linear growth rate of matter fluctuations. Noticethat the parameter combination fσ8 is insensitive to differencesbetween the clustering of galaxies and dark matter, i.e., to galaxybias (Song & Percival 2009). In the base ΛCDM cosmology, thegrowth factor f (z) is well approximated as f (z) = Ωm(z)0.545.More directly, in linear theory the quadrupole of the redshift-space clustering anisotropy actually probes the density-velocitycorrelation power spectrum, and we therefore define

fσ8(z) ≡

[σ(vd)

8 (z)]2

σ(dd)8 (z)

, (33)

as an approximate proxy for the quantity actually being mea-sured. Here σ(vd)

8 measures the smoothed density-velocity corre-lation and is defined analogously toσ8 ≡ σ

(dd)8 , but using the cor-

relation power spectrum Pvd(k), where v = −∇ · vN/H and vN isthe Newtonian-gauge (peculiar) velocity of the baryons and darkmatter, and d is the total matter density perturbation. This defi-nition assumes that the observed galaxies follow the flow of thecold matter, not including massive neutrino velocity effects. Formodels close to ΛCDM, where the growth is nearly scale inde-pendent, it is equivalent to defining fσ8 in terms of the growth ofthe baryon+CDM density perturbations (excluding neutrinos).

The use of RSD as a measure of the growth of structure isstill under active development and is considerably more difficultthan measuring the positions of BAO features. Firstly, adopt-ing the wrong fiducial cosmology can induce an anisotropy inthe clustering of galaxies, via the Alcock-Paczynski (AP) ef-fect, which is strongly degenerate with the anisotropy inducedby peculiar motions. Secondly, much of the RSD signal cur-rently comes from scales where nonlinear effects and galaxybias are significant and must be accurately modelled in order torelate the density and velocity fields (see, e.g., the discussionsin Bianchi et al. 2012; Okumura et al. 2012; Reid et al. 2014;White et al. 2015).

27


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

z

0.2

0.3

0.4

0.5

0.6

0.7

fσ

8

6DFGS

SDSS MGSSDSS LRG

BOSS LOWZBOSS CMASS

VIPERS

WiggleZ

Fig. 16. Constraints on the growth rate of fluctuations fromvarious redshift surveys in the base ΛCDM model: green star(6dFGRS, Beutler et al. 2012); purple square (SDSS MGS,Howlett et al. 2015); cyan cross (SDSS LRG, Oka et al. 2014);red triangle (BOSS LOWZ survey, Gil-Marın et al. 2015); largered circle (BOSS CMASS, as analysed by Samushia et al. 2014);blue circles (WiggleZ, Blake et al. 2012); and green diamond(VIPERS, de la Torre et al. 2013). The points with dashed rederror bars correspond to alternative analyses of BOSS CMASSfrom Beutler et al. (2014, small circle, offset for clarity) andChuang et al. (2013, small square). Of the BOSS CMASSpoints, two are based on the same DR11 data set (Samushia et al.2014; Beutler et al. 2014), while the third is based on the morerecent DR12 (Chuang et al. 2013), and are therefore not inde-pendent. The grey bands show the range allowed by PlanckTT+lowP+lensing in the base ΛCDM model. Where available(for SDSS MGS and BOSS), we have plotted conditional con-straints on fσ8 assuming a Planck ΛCDM background cosmol-ogy. The WiggleZ points are plotted conditional on the meanPlanck cosmology prediction for FAP (evaluated using the co-variance between fσ8 and FAP given in Blake et al. 2012). The6dFGS point is at sufficiently low redshift that it is insensitive tothe cosmology.

Current constraints,24 assuming a Planck base ΛCDMmodel, are shown in Fig. 16. Neglecting the AP effect can lead tobiased measurements of fσ8 if the assumed cosmology differs,and to significant underestimation of the errors (Howlett et al.2015). The analyses summarized in Fig. 16 solve simultaneouslyfor RSD and the AP effect, except for the 6dFGS point (whichis insensitive to cosmology) and the VIPERS point (which has alarge error). The grey bands show the range allowed by PlanckTT+lowP+lensing in the base ΛCDM model, and are consistentwith the RSD data. The tightest constraints on fσ8 in this figurecome from the BOSS CMASS-DR11 analyses of Beutler et al.(2014) and Samushia et al. (2014). The Beutler et al. (2014)analysis is performed in Fourier space and shows a small biasin fσ8 compared to numerical simulations when fitting overthe wavenumber range 0.01–0.20 hMpc−1. The Samushia et al.

24The constraint of Chuang et al. (2013) plotted in the original ver-sion of this paper was subsequently shown to be in error. We there-fore now show updated BOSS data points for DR12 from Chuang et al.(2013, for CMASS) and Gil-Marın et al. (2015, for LOWZ).

0.60 0.65 0.70 0.75

FAP(0.57)

0.32

0.40

0.48

0.56

fσ

8(0

.57

)

BOSS CMASS (Samushia et al.)

BOSS CMASS (Beutler et al.)

Planck TT+lowP+lensing

Fig. 17. 68 % and 95 % contours in the fσ8–FAP plane(marginalizing over Dv/rs) for the CMASS-DR11 sample asanalysed by Samushia et al. (2014, solid, our defult), andBeutler et al. (2014, dotted). The green contours show the con-straint from Planck TT+lowP+lensing in the base ΛCDMmodel.

(2014) analysis is performed in configuration space and showsno evidence of biases when compared to numerical simulations.The updated DR12 CMASS result from Chuang et al. (2013)marginalizes over a polynomial model for systematic errors inthe correlation function monopole, and is consistent with theseand the Planck constraints, with a somewhat larger error bar.

The Samushia et al. (2014) results are expressed as a 3 × 3covariance matrix for the three parameters DV/rdrag, FAP andfσ8, evaluated at an effective redshift of zeff = 0.57, where FAPis the “Alcock-Paczynski” parameter

FAP(z) = (1 + z)DAH(z)

c. (34)

The principal degeneracy is between fσ8 and FAP and is il-lustrated in Fig. 17, compared to the constraint from PlanckTT+lowP+lensing for the base ΛCDM cosmology. The Planckresults sit slightly high but overlap the 68 % contour fromSamushia et al. (2014). The Planck result lies about 1.5σ higherthan the Beutler et al. (2014) analysis of the BOSS CMASS sam-ple.

RSD measurements are not used in combination with Planckin this paper. However, in the companion paper exploring darkenergy and modified gravity (Planck Collaboration XIV 2016),the RSD/BAO measurements of Samushia et al. (2014) are usedtogether with Planck. Where this is done, we exclude theAnderson et al. (2014) BOSS-CMASS results from the BAOlikelihood. Since Samushia et al. (2014) do not apply a densityfield reconstruction in their analysis, the BAO constraints fromBOSS-CMASS are then slightly weaker, though consistent, withthose of Anderson et al. (2014).

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5.5.2. Weak gravitational lensing

Weak gravitational lensing offers a potentially powerful tech-nique for measuring the amplitude of the matter fluctuation spec-trum at low redshifts. Currently, the largest weak lensing dataset is provided by the CFHTLenS survey (Heymans et al. 2012;Erben et al. 2013). The first science results from this survey ap-peared shortly before the completion of PCP13 and it was notpossible to do much more than offer a cursory comparison withthe Planck 2013 results. As reported in PCP13, at face valuethe results from CFHTLenS appeared to be in tension with thePlanck 2013 base ΛCDM cosmology at about the 2–3σ level.Since neither the CFHTLenS results nor the 2015 Planck resultshave changed significantly from those in PCP13, it is worth dis-cussing this discrepancy in more detail in this paper.

Weak lensing data can be analysed in various ways. For ex-ample, one can compute two correlation functions from the ellip-ticities of pairs of images separated by angle θ,which are relatedto the convergence power spectrum Pκ(`) of the survey at multi-pole ` via

ξ±(θ) =1

2π

∫d``Pκ(`)J±(`θ), (35)

where the Bessel functions in (35) are J+ ≡ J0 and J− ≡ J4(see, e.g., Bartelmann & Schneider 2001). Much of the informa-tion from the CFHTLenS survey correlation function analysescomes from wavenumbers at which the matter power spectrumis strongly nonlinear, complicating any direct comparison withPlanck.

This can be circumventing by performing a 3D spherical har-monic analysis of the shear field, allowing one to impose lowerlimits on the wavenumbers that contribute to a weak lensing like-lihood. This has been done by Kitching et al. (2014). Includingonly wavenumbers with k ≤ 1.5 hMpc−1, Kitching et al. (2014)find constraints in the σ8–Ωm plane that are consistent with theresults from Planck. However, by excluding modes with higherwavenumbers, the lensing constraints are weakened. When theyincrease the wavenumber cut-off to k = 5 hMpc−1 some tensionwith Planck begins to emerge (which these authors argue maybe an indication of the effects of baryonic feedback in suppress-ing the matter power spectrum at small scales). The large-scaleproperties of CFHTLenS therefore seem broadly consistent withPlanck and it is only as CFHTLenS probes higher wavenumbers,particular in the 2D and tomographic correlation function anal-yses (Heymans et al. 2013; Kilbinger et al. 2013; Fu et al. 2014;MacCrann et al. 2015), that apparently strong discrepancies withPlanck appear.

The situation is summarized in Fig. 18. The sample pointsshow parameter values in the σ8–Ωm plane for the ΛCDM basemodel, computed from the Heymans et al. (2013, hereafter H13)tomographic measurements of ξ±. These data consist of correla-tion function measurements in six photometric redshift bins ex-tending over the redshift range 0.2–1.3. We use the blue galaxysample, since H13 find that this sample shows no evidence forintrinsic galaxy alignments (simplifying the comparison withtheory) and we apply the “conservative” cuts of H13, intendedto reduce sensitivity to the nonlinear part of the power spec-trum; these cuts eliminate measurements with θ < 3′ for anyredshift combination that involves the lowest two redshift bins.Here we have used the halofit prescription of Takahashi et al.(2012) to model the nonlinear power spectrum, but do not in-clude any model of baryon feedback or intrinsic alignments.For the lensing-only constraint we also impose additional pri-ors in a similar way to the CMB lensing analysis describedin Planck Collaboration XV (2016), i.e., Gaussian priors Ωbh2 =

0.2 0.3 0.4 0.5 0.6

Ωm

0.6

0.8

1.0

σ8

WL+BAO

WL+θMC+BAO

Planck TT+lowP

32

40

48

56

64

72

80

88

96

H0

Fig. 18. Samples in the σ8–Ωm plane from the H13 CFHTLenSdata (with angular cuts as discussed in the text), coloured by thevalue of the Hubble parameter, compared to the joint constraintswhen the lensing data are combined with BAO (blue), and BAOwith the CMB acoustic scale parameter fixed to θMC = 1.0408(green). For comparison, the Planck TT+lowP constraint con-tours are shown in black. The grey bands show the constraintfrom Planck CMB lensing. We impose a weak prior on the pri-moridal amplitude, 2 < ln(1010As) < 4, which has some impacton the distribution of CFHTLenS-only samples.

0.0223 ± 0.0009 and ns = 0.96 ± 0.02, where the exact val-ues (chosen to span reasonable ranges given CMB data) havelittle impact on the results. The sample range shown also re-stricts the Hubble parameter to 0.2 < h < 1; note that whencomparing with constraint contours, the location of the contourscan change significantly depending on the H0 prior range as-sumed. We also use a weak prior on the primoridal amplitude,2 < ln(1010As) < 4, which shows up the strong correlation be-tween Ωm–σ8–H0 in the region of parameter space relevant forcomparison with Planck. In Fig. 18 we only show lensing con-tours after the samples have been projected into the space al-lowed by the BAO data (blue contours), or also additionally re-stricting to the reduced space where θMC is fixed to the Planckvalue, which is accurately measured. The black contours showthe constraints from Planck TT+lowP.

The lensing samples just overlap with Planck, and superfi-cially one might conclude that the two data sets are consistent.However, the weak lensing constraints approximately define a1D degeneracy in the 3D Ωm–σ8–H0 space, so consistency of theHubble parameter at each point in the projected space must alsobe considered (see appendix E1 of Planck Collaboration XV2016). Comparing the contours in Fig. 18 (the regions wherethe weak lensing constraints are consistent with BAO obser-vations) the CFHTLenS data favour a lower value of σ8 thanthe Planck data (and much of the area of the blue contoursalso has higher Ωm). However, even with the conservative an-gular cuts applied by H13, the weak lensing constraints de-pend on the nonlinear model of the power spectrum and on thepossible influence of baryonic feedback in reshaping the mat-ter power spectrum at small spatial scales (Harnois-Deraps et al.2015; MacCrann et al. 2015). The importance of these effectscan be reduced by imposing even more conservative angular

29


cuts on ξ±, but of course, this weakens the statistical powerof the weak lensing data. The CFHTLenS data are not usedin combination with Planck in this paper (apart from specificcases in Sects. 6.3 and 6.4.4) and, in any case, would have lit-tle impact on most of the extended ΛCDM constraints discussedin Sect. 6. Weak lensing can, however, provide important con-straints on dark energy and modified gravity. The CFHTLenSdata are therefore used in combination with Planck in the com-panion paper (Planck Collaboration XIV 2016), which exploresseveral halofit prescriptions and the impact of applying moreconservative angular cuts to the H13 measurements.

5.5.3. Planck cluster counts

In 2013 we noted a possible tension between our primary CMBconstraints and those from the Planck SZ cluster counts, with theclusters preferring lower values of σ8 in the base ΛCDM modelin some analyses (Planck Collaboration XX 2014). The compar-ison is interesting because the cluster counts directly measure σ8at low redshift; any tension could signal the need for extensionsto the base model, such as non-minimal neutrino mass (thoughsee Sect. 6.4). However, limited knowledge of the scaling rela-tion between SZ signal and mass have hampered the interpreta-tion of this result.

With the full mission data we have created a larger cata-logue of SZ clusters with a more accurate characterization ofits completeness (Planck Collaboration XXIV 2016). By fittingthe counts in redshift and signal-to-noise, we are able to si-multaneously constrain the slope of the SZ signal–mass scal-ing relation and the cosmological parameters. A major uncer-tainty, however, remains the overall mass calibration, which inPlanck Collaboration XX (2014) we quantified with a “hydro-static bias” parameter, (1 − b), with a fiducial value of 0.8 anda range 0.7 < (1 − b) < 1 (consistent with some other stud-ies, e.g., Simet et al. 2015). In the base ΛCDM model, the pri-mary CMB constraints prefer a normalization below the lowerend of this range, (1 − b) ≈ 0.6. The recent, empirical nor-malization of the relation by the Weighing the Giants lensingprogramme (WtG; von der Linden et al. 2014) gives 0.69± 0.07for the 22 clusters in common with the Planck cluster sample.This calibration reduces the tension with the primary CMB con-straints in base ΛCDM. In contrast, correlating the entire Planck2015 SZ cosmology sample with Planck CMB lensing gives1/(1−b) = 1.0±0.2 (Planck Collaboration XXIV 2016), towardthe upper end of the range adopted in Planck Collaboration XX(2014), although with a large uncertainty. An alternative lens-ing calibration analysis by the Canadian Cluster ComparisonProject, which uses 37 clusters in common with the Planckcluster sample (Hoekstra et al. 2015) finds (1 − b) = 0.76 ±0.05 (stat.)± 0.06 (syst.), which lies between the other two masscalibrations. These calibrations are not yet definitive and the sit-uation will continue to evolve with improvements in mass mea-surements from larger samples of clusters.

A recent analysis of cluster counts for an X-ray-selectedsample (REFLEX II) shows some tension with the Planck baseΛCDM cosmology (Bohringer et al. 2014). However, an analy-sis of cluster counts of X-ray-selected clusters by the WtG col-laboration, incorporating the WtG weak lensing mass calibra-tion, finds σ8(Ωm/0.3)0.17 = 0.81±0.03, in good agreement withthe Planck CMB results for base ΛCDM (Mantz et al. 2015).This raises the possibility that there may be systematic biasesin the assumed scaling relations for SZ-selected clusters com-pared to X-ray-selected clusters (in addition to a possible masscalibration bias). Mantz et al. (2015) give a brief review of re-

0.27 0.30 0.33 0.36

Ωm

0.75

0.80

0.85

0.90

0.95

σ8


Planck TT,TE,EE+lowP+lensing

Planck TT,TE,EE+reion prior

Planck TT

Planck TT,TE,EE

+lensing

+BAO

+zre > 6.5

Fig. 19. Marginalized constraints on parameters of the baseΛCDM model without low-` E-mode polarization (filled con-tours), compared to the constraints from using low-` E-modepolarization (unfilled contours) or assuming a strong prior thatreionization was at zre = 7 ± 1 and zre > 6.5 (“reion prior,”dashed contours). Grey bands show the constraint from CMBlensing alone.

cent determinations of σ8 from X-ray, optically-selected, andSZ-selected samples, to which we refer the reader. More detaileddiscussion of constraints from combining Planck cluster countswith primary CMB anisotropies and other data sets can be foundin Planck Collaboration XXIV (2016).

5.6. Cosmic concordance?

Table 4 summarizes the cosmological parameters in the baseΛCDM for Planck combined with various data sets discussed inthis section. Although we have seen from the survey presentedabove that base ΛCDM is consistent with a wide range of cos-mological data, there are two areas of tension:

1. the Lyα BAO measurements at high redshift (Sect. 5.2);2. the Planck CMB estimate of the amplitude of the fluctuation

spectrum and the lower values inferred from weak lensing,and (possibly) cluster counts and redshift space distortions(Sect. 5.5).

The first point to note is that the astrophysical data in areas(1) and (2) are complex and more difficult to interpret than mostof the astrophysical data sets discussed in this section. The inter-pretation of the data in area (2) depends on nonlinear modellingof the power spectrum, and in the case of clusters and weak lens-ing, on uncertain baryonic physics. Understanding these effectsmore accurately sets a direction for future research.

It is, however, worth reviewing our findings on σ8 and Ωmfrom Planck assuming base ΛCDM. These are summarized in

30


Fig. 19 and the following constraints:

σ8 = 0.829 ± 0.014, Planck TT+lowP; (36a)σ8 = 0.815 ± 0.009, Planck TT+lowP+lensing; (36b)σ8 = 0.810 ± 0.006, Planck TT+lensing+zre. (36c)

The last line imposes a Gaussian prior of zre = 7 ± 1 with alimit zre > 6.5 on the reionization redshift in place of the reion-ization constraints from the lowP likelihood. As discussed inSect. 3.4, such a low redshift of reionization is close to the low-est plausible value allowed by astrophysical data (though suchlow values are not favoured by either the WMAP or LFI polar-ization data). The addition of Planck lensing data pulls σ8 downby about 1σ from the Planck TT+lowP value, so Eq. (36c) isthe lowest possible range allowed by the Planck CMB data. Asshown in Fig. 19, adding the T E and EE spectra at high mul-tipoles does not change the Planck constraints. If a convincingcase can be made that astrophysical data conflict with the esti-mate of Eq. (36c), then this will be powerful evidence for newphysics beyond base ΛCDM with minimal-mass neutrinos.

A number of authors have interpreted the discrepancies inarea (2) as evidence for new physics in the neutrino sector (e.g.,Planck Collaboration XX 2014; Hamann & Hasenkamp 2013;Battye & Moss 2014; Battye et al. 2015; Wyman et al. 2014;Beutler et al. 2014). They use various data combinations to-gether with Planck to argue for massive neutrinos with mass∑

mν ≈ 0.3 eV or for a single sterile neutrino with somewhathigher mass. The problem here is that any evidence for newneutrino physics is driven mainly by the additional astrophysi-cal data, not by Planck CMB anisotropy measurements. In addi-tion, the external data sets are not entirely consistent, so tensionsremain. As discussed in PCP13 (see also Leistedt et al. 2014;Battye et al. 2015) Planck data usually favour base ΛCDM overextended models. Implications of the Planck 2015 data for neu-trino physics are discussed in Sect. 6.4 and tensions betweenPlanck and external data in various extended neutrino modelsare discussed further in Sect. 6.4.4.

As mentioned above, we do not use RSD or galaxy weaklensing measurements for combined constraints in this paper(apart from Sects. 6.3 and 6.4.4, where we use the CFHTLenSdata) . They are, however, used in the paper exploring constraintson dark energy and modified gravity (Planck Collaboration XIV2016). For some models discussed in that paper, the combinationof Planck, RSD, and weak lensing data does prefer extensions tothe base ΛCDM cosmology.

6. Extensions to the base ΛCDM model

6.1. Grid of models

The full grid results are available online.25 Figure 20 and Table 5summarize the constraints on one-parameter extensions to baseΛCDM. As in PCP13, we find no strong evidence in favour ofany of these simple one-parameter extensions using Planck orPlanck combined with BAO. The entire grid has been run usingboth the Plik and CamSpec likelihoods. As noted in Sect. 3, theparameters derived from these two TT likelihoods agree to betterthan 0.5σ for base ΛCDM. This level of agreement also holdsfor the extended models analysed in our grid. In Sect. 3 we alsopointed out that we have definite evidence, by comparing spec-tra computed with different frequency combinations, of residual

25See the Planck Legacy Archive, http://www.cosmos.esa.int/web/planck/pla, which contains considerably more detailed informa-tion than presented in this paper.

systematics in the T E and EE spectra. These systematics aver-age down in the coadded T E and EE spectra, but the remaininglevel of systematics in these coadded spectra are not yet wellquantified (though they are small). Thus, we urge the reader totreat parameters computed from the TT,TE,EE likelihoods withsome caution. In the case of polarization, the agreement betweenthe Plik and CamSpec T E and EE likelihoods is less good, withshifts in parameters of up to 1.5σ (though such large shifts areunusual). In general, the behaviour of the TT,TE,EE likelihoodsis as shown in Fig. 20. For extended models, the addition of thePlanck polarization data at high multipoles reduces the errors onextended parameters compared to the Planck temperature dataand pulls the parameters towards those of base ΛCDM. A sim-ilar behaviour is seen if the Planck TT (or Planck TT,TE,EE)data are combined with BAO.

The rest of this section discusses the grid results in more de-tail and also reports results on some additional models (specifi-cally dark matter annihilation, tests of the recombination history,and cosmic defects) that are not included in our grid.

6.2. Early-Universe physics

Arguably the most important result from 2013 Planck analysiswas the finding that simple single-field inflationary models, witha tilted scalar spectrum ns ≈ 0.96, provide a very good fit tothe Planck data. We found no evidence for a tensor componentor running of the scalar spectral index, no strong evidence forisocurvature perturbations or features in the primordial powerspectrum (Planck Collaboration XXII 2014), and no evidencefor non-Gaussianity (Planck Collaboration XXIV 2014), cosmicstrings or other topological defects (Planck Collaboration XXV2014). On large angular scales, the Planck data showed someevidence for “anomalies” seen previously in the WMAP data(Bennett et al. 2011). These include a dip in the power spectrumin the multipole range 20 <∼ ` <∼ 30 (see Fig. 1) and some evi-dence for a departure from statistical isotropy on large angularscales (Planck Collaboration XXIII 2014). However, the statisti-cal significance of these anomalies is not high enough to providecompelling evidence for new physics beyond simple single-fieldinflation.

The Planck 2013 results led to renewed interest in the R2 in-flationary model, originally introduced by Starobinsky (1980),and related inflationary models that have flat effective poten-tials of similar form (e.g., Kallosh & Linde 2013; Ferrara et al.2013; Buchmuller et al. 2013; Ellis et al. 2013). A characteristicof these models is that they produce a red tilted scalar spectrumand a low tensor-to-scalar ratio. For reference, the Starobinskymodel predicts

ns ≈ 1 −2N∈ (0.960, 0.967), (37a)

r ≈12N2 ∈ (0.003, 0.005), (37b)

dns

d ln k≈ −

2N2 ∈ (−0.0008,−0.0006), (37c)

where N is the number of e-foldings between the end of in-flation and the time that our present day Hubble scale crossedthe inflationary horizon, and numerical values are for the range50 ≤ N ≤ 60.

Although the Planck 2013 results stimulated theoreticalwork on inflationary models with low tensor-to-scalar ratios, thecosmological landscape became more complicated following thedetection of a B-mode polarization anisotropy by the BICEP2

31

http://www.cosmos.esa.int/web/planck/pla

http://www.cosmos.esa.int/web/planck/pla


Table 4. Parameter 68 % confidence limits for the base ΛCDM model from Planck CMB power spectra, in combination with lensingreconstruction (“lensing”) and external data (“ext”, BAO+JLA+H0). While we see no evidence that systematic effects in polarizationare biasing parameters in the base ΛCDM model, a conservative choice would be to use the parameter values listed in Column 3(i.e., for TT+lowP+lensing). Nuisance parameters are not listed here for brevity, but can be found in the extensive tables on thePlanck Legacy Archive, http://pla.esac.esa.int/pla; however, the last three parameters listed here give a summary measureof the total foreground amplitude (in µK2) at ` = 2000 for the three high-` temperature power spectra used by the likelihood.In all cases the helium mass fraction used is predicted by BBN from the baryon abundance (posterior mean YP ≈ 0.2453, withtheoretical uncertainties in the BBN predictions dominating over the Planck error on Ωbh2). The Hubble constant is given in unitsof km s−1 Mpc−1, while r∗ is in Mpc and wavenumbers are in Mpc−1.

TT+lowP TT+lowP+lensing TT+lowP+lensing+ext TT,TE,EE+lowP TT,TE,EE+lowP+lensing TT,TE,EE+lowP+lensing+extParameter 68 % limits 68 % limits 68 % limits 68 % limits 68 % limits 68 % limits

Ωbh2 . . . . . . . . . . . 0.02222 ± 0.00023 0.02226 ± 0.00023 0.02227 ± 0.00020 0.02225 ± 0.00016 0.02226 ± 0.00016 0.02230 ± 0.00014

Ωch2 . . . . . . . . . . . 0.1197 ± 0.0022 0.1186 ± 0.0020 0.1184 ± 0.0012 0.1198 ± 0.0015 0.1193 ± 0.0014 0.1188 ± 0.0010

100θMC . . . . . . . . . 1.04085 ± 0.00047 1.04103 ± 0.00046 1.04106 ± 0.00041 1.04077 ± 0.00032 1.04087 ± 0.00032 1.04093 ± 0.00030

τ . . . . . . . . . . . . . 0.078 ± 0.019 0.066 ± 0.016 0.067 ± 0.013 0.079 ± 0.017 0.063 ± 0.014 0.066 ± 0.012

ln(1010As) . . . . . . . . 3.089 ± 0.036 3.062 ± 0.029 3.064 ± 0.024 3.094 ± 0.034 3.059 ± 0.025 3.064 ± 0.023

ns . . . . . . . . . . . . 0.9655 ± 0.0062 0.9677 ± 0.0060 0.9681 ± 0.0044 0.9645 ± 0.0049 0.9653 ± 0.0048 0.9667 ± 0.0040

H0 . . . . . . . . . . . . 67.31 ± 0.96 67.81 ± 0.92 67.90 ± 0.55 67.27 ± 0.66 67.51 ± 0.64 67.74 ± 0.46

ΩΛ . . . . . . . . . . . . 0.685 ± 0.013 0.692 ± 0.012 0.6935 ± 0.0072 0.6844 ± 0.0091 0.6879 ± 0.0087 0.6911 ± 0.0062

Ωm . . . . . . . . . . . . 0.315 ± 0.013 0.308 ± 0.012 0.3065 ± 0.0072 0.3156 ± 0.0091 0.3121 ± 0.0087 0.3089 ± 0.0062

Ωmh2 . . . . . . . . . . 0.1426 ± 0.0020 0.1415 ± 0.0019 0.1413 ± 0.0011 0.1427 ± 0.0014 0.1422 ± 0.0013 0.14170 ± 0.00097

Ωmh3 . . . . . . . . . . 0.09597 ± 0.00045 0.09591 ± 0.00045 0.09593 ± 0.00045 0.09601 ± 0.00029 0.09596 ± 0.00030 0.09598 ± 0.00029

σ8 . . . . . . . . . . . . 0.829 ± 0.014 0.8149 ± 0.0093 0.8154 ± 0.0090 0.831 ± 0.013 0.8150 ± 0.0087 0.8159 ± 0.0086

σ8Ω0.5m . . . . . . . . . . 0.466 ± 0.013 0.4521 ± 0.0088 0.4514 ± 0.0066 0.4668 ± 0.0098 0.4553 ± 0.0068 0.4535 ± 0.0059

σ8Ω0.25m . . . . . . . . . 0.621 ± 0.013 0.6069 ± 0.0076 0.6066 ± 0.0070 0.623 ± 0.011 0.6091 ± 0.0067 0.6083 ± 0.0066

zre . . . . . . . . . . . . 9.9+1.8−1.6 8.8+1.7

−1.4 8.9+1.3−1.2 10.0+1.7

−1.5 8.5+1.4−1.2 8.8+1.2

−1.1

109As . . . . . . . . . . 2.198+0.076−0.085 2.139 ± 0.063 2.143 ± 0.051 2.207 ± 0.074 2.130 ± 0.053 2.142 ± 0.049

109Ase−2τ . . . . . . . . 1.880 ± 0.014 1.874 ± 0.013 1.873 ± 0.011 1.882 ± 0.012 1.878 ± 0.011 1.876 ± 0.011

Age/Gyr . . . . . . . . 13.813 ± 0.038 13.799 ± 0.038 13.796 ± 0.029 13.813 ± 0.026 13.807 ± 0.026 13.799 ± 0.021

z∗ . . . . . . . . . . . . 1090.09 ± 0.42 1089.94 ± 0.42 1089.90 ± 0.30 1090.06 ± 0.30 1090.00 ± 0.29 1089.90 ± 0.23

r∗ . . . . . . . . . . . . 144.61 ± 0.49 144.89 ± 0.44 144.93 ± 0.30 144.57 ± 0.32 144.71 ± 0.31 144.81 ± 0.24

100θ∗ . . . . . . . . . . 1.04105 ± 0.00046 1.04122 ± 0.00045 1.04126 ± 0.00041 1.04096 ± 0.00032 1.04106 ± 0.00031 1.04112 ± 0.00029

zdrag . . . . . . . . . . . 1059.57 ± 0.46 1059.57 ± 0.47 1059.60 ± 0.44 1059.65 ± 0.31 1059.62 ± 0.31 1059.68 ± 0.29

rdrag . . . . . . . . . . . 147.33 ± 0.49 147.60 ± 0.43 147.63 ± 0.32 147.27 ± 0.31 147.41 ± 0.30 147.50 ± 0.24

kD . . . . . . . . . . . . 0.14050 ± 0.00052 0.14024 ± 0.00047 0.14022 ± 0.00042 0.14059 ± 0.00032 0.14044 ± 0.00032 0.14038 ± 0.00029

zeq . . . . . . . . . . . . 3393 ± 49 3365 ± 44 3361 ± 27 3395 ± 33 3382 ± 32 3371 ± 23

keq . . . . . . . . . . . . 0.01035 ± 0.00015 0.01027 ± 0.00014 0.010258 ± 0.000083 0.01036 ± 0.00010 0.010322 ± 0.000096 0.010288 ± 0.000071

100θs,eq . . . . . . . . . 0.4502 ± 0.0047 0.4529 ± 0.0044 0.4533 ± 0.0026 0.4499 ± 0.0032 0.4512 ± 0.0031 0.4523 ± 0.0023

f 1432000 . . . . . . . . . . . 29.9 ± 2.9 30.4 ± 2.9 30.3 ± 2.8 29.5 ± 2.7 30.2 ± 2.7 30.0 ± 2.7

f 143×2172000 . . . . . . . . . 32.4 ± 2.1 32.8 ± 2.1 32.7 ± 2.0 32.2 ± 1.9 32.8 ± 1.9 32.6 ± 1.9

f 2172000 . . . . . . . . . . . 106.0 ± 2.0 106.3 ± 2.0 106.2 ± 2.0 105.8 ± 1.9 106.2 ± 1.9 106.1 ± 1.8

Table 5. Constraints on 1-parameter extensions to the base ΛCDM model for combinations of Planck power spectra, Planck lensing,and external data (BAO+JLA+H0, denoted “ext”). All limits and confidence regions quoted here are 95 %.

Parameter TT TT+lensing TT+lensing+ext TT,TE,EE TT,TE,EE+lensing TT,TE,EE+lensing+ext

ΩK . . . . . . . . . . . . . . −0.052+0.049−0.055 −0.005+0.016

−0.017 −0.0001+0.0054−0.0052 −0.040+0.038

−0.041 −0.004+0.015−0.015 0.0008+0.0040

−0.0039Σmν [eV] . . . . . . . . . . < 0.715 < 0.675 < 0.234 < 0.492 < 0.589 < 0.194Neff . . . . . . . . . . . . . . 3.13+0.64

−0.63 3.13+0.62−0.61 3.15+0.41

−0.40 2.99+0.41−0.39 2.94+0.38

−0.38 3.04+0.33−0.33

YP . . . . . . . . . . . . . . . 0.252+0.041−0.042 0.251+0.040

−0.039 0.251+0.035−0.036 0.250+0.026

−0.027 0.247+0.026−0.027 0.249+0.025

−0.026dns/d ln k . . . . . . . . . . −0.008+0.016

−0.016 −0.003+0.015−0.015 −0.003+0.015

−0.014 −0.006+0.014−0.014 −0.002+0.013

−0.013 −0.002+0.013−0.013

r0.002 . . . . . . . . . . . . . < 0.103 < 0.114 < 0.114 < 0.0987 < 0.112 < 0.113w . . . . . . . . . . . . . . . −1.54+0.62

−0.50 −1.41+0.64−0.56 −1.006+0.085

−0.091 −1.55+0.58−0.48 −1.42+0.62

−0.56 −1.019+0.075−0.080

32

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−0.18

−0.12

−0.06

0.00

ΩK

0.4

0.8

1.2

1.6

Σmν

[eV

]

2.4

3.0

3.6

4.2

Neff

0.20

0.24

0.28

0.32

YP

−0.030

−0.015

0.000

0.015

dn

s/d

lnk

0.0216 0.0224 0.0232

Ωbh2

0.08

0.16

0.24

0.32

r 0.0

02

0.112 0.120 0.128

Ωch20.93 0.96 0.99 1.02

ns

40 50 60 70 80

H0

0.64 0.72 0.80 0.88

σ8

Planck TT+lowP Planck TT,TE,EE+lowP Planck TT,TE,EE+lowP+BAO

Fig. 20. 68 % and 95 % confidence regions on 1-parameter extensions of the base ΛCDM model for Planck TT+lowP (grey),Planck TT,TE,EE+lowP (red), and Planck TT,TE,EE+lowP+BAO (blue). Horizontal dashed lines correspond to the parametervalues assumed in the base ΛCDM cosmology, while vertical dashed lines show the mean posterior values in the base model forPlanck TT,TE,EE+lowP+BAO.

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φ2

0.95 0.96 0.97 0.98 0.99 1.00

ns

0.00

0.05

0.10

0.15

0.20

0.25r 0

.00

2

N=

50

N=

60

ConvexConcave

φ

Planck TT+lowP+lensing (∆Neff = 0.39)

ΛCDM Planck TT+lowP

+lensing+ext

φ2

0.95 0.96 0.97 0.98 0.99 1.00

ns

0.00

0.05

0.10

0.15

0.20

0.25

r 0.0

02

N=

50

N=

60

ConvexConcave

φ

Planck TT+lowP

Planck TT+lowP+BKP

+lensing+ext

Fig. 21. Left: Constraints on the tensor-to-scalar ratio r0.002 in the ΛCDM model, using Planck TT+lowP and PlanckTT+lowP+lensing+BAO+JLA+H0 (red and blue, respectively) assuming negligible running and the inflationary consistency rela-tion. The result is model-dependent; for example, the grey contours show how the results change if there were additional relativisticdegrees of freedom with ∆Neff = 0.39 (disfavoured, but not excluded, by Planck). Dotted lines show loci of approximately con-stant e-folding number N, assuming simple V ∝ (φ/mPl)p single-field inflation. Solid lines show the approximate ns–r relation forquadratic and linear potentials, to first order in slow roll; red lines show the approximate allowed range assuming 50 < N < 60 anda power-law potential for the duration of inflation. The solid black line (corresponding to a linear potential) separates concave andconvex potentials. Right: Equivalent constraints in the ΛCDM model when adding B-mode polarization results corresponding to thedefault configuration of the BICEP2/Keck Array+Planck (BKP) likelihood. These exclude the quadratic potential at a higher levelof significance compared to the Planck-alone constraints.

team (BICEP2 Collaboration 2014). If the BICEP2 signal wereprimarily caused by primordial gravitational waves, then the in-ferred tensor-to-scalar ratio would have been r0.01 ≈ 0.2,26 ap-parently in conflict with the 2013 Planck 95 % upper limit ofr0.002 < 0.11, based on fits to the temperature power spectrum.Since the Planck constraints on r are highly model dependent(and fixed mainly by lower k) it is possible to reconcile these re-sults by introducing additional parameters, such as large tilts orstrong running of the spectral indices.

The situation has been clarified following a joint analysisof BICEP2/Keck observations and Planck polarization data re-ported in BKP. This analysis shows that polarized dust emissioncontributes a significant part of the BICEP2 signal. Correctingfor polarized dust emission, BKP report a 95 % upper limit ofr0.05 < 0.12 on scale-invariant tensor modes, eliminating thetension between the BICEP2 and the Planck 2013 results. Thereis therefore no evidence for inflationary tensor modes from B-mode polarization measurements at this time (although the BKPanalysis leaves open the possibility of a much higher tensor-to-scalar ratio than the prediction of Eq. 37b for Starobinsky-typemodels).

The layout of the rest of this subsection is as follows. InSect. 6.2.1 we review the Planck 2015 and Planck+BKP con-straints on ns and r. Constraints on the running of the scalar spec-tral index are presented in Sect. 6.2.2. Polarization data providea powerful way of testing for isocurvature modes, as discussedin Sect. 6.2.3. Finally, Sect. 6.2.4 summarizes our results on spa-

26The pivot scale quoted here is roughly appropriate for the multi-poles probed by BICEP2.

tial curvature. A discussion of specific inflationary models andtests for features in the primordial power spectrum can be foundin Planck Collaboration XX (2016).

6.2.1. Scalar spectral index and tensor fluctuations

Primordial tensor fluctuations (gravitational waves) contributeto both the CMB temperature and polarization power spectra.Gravitational waves entering the horizon between recombina-tion and the present day generate a tensor contribution to thelarge-scale CMB temperature anisotropy. In this data release,the strongest constraint on tensor modes from Planck data stillcomes from the CMB temperature spectrum at ` <∼ 100. The cor-responding comoving wavenumbers probed by the Planck tem-perature spectrum have k <∼ 0.008 Mpc−1, with very little sensi-tivity to higher wavenumbers because gravitational waves decayon sub-horizon scales. The precision of the Planck constraint islimited by cosmic variance of the large-scale anisotropies (whichare dominated by the scalar component), and it is also model de-pendent. In polarization, in addition to B-modes, the EE and T Espectra also contain a signal from tensor modes coming from thelast-scattering and reionization epochs. However, in this releasethe addition of Planck polarization constraints at ` ≥ 30 do notsignificantly change the results from temperature and low-` po-larization (see Table 5).

Figure 21 shows the 2015 Planck constraint in the ns–r plane,adding r as a 1-parameter extension to base ΛCDM. For baseΛCDM (r = 0), the value of ns is

ns = 0.9655 ± 0.0062, Planck TT+lowP. (38)

34


We highlight this number here since ns, a key parameter for in-flationary cosmology, shows one of the largest shifts of any pa-rameter in base ΛCDM between the Planck 2013 and Planck2015 analyses (about 0.7σ). As explained in Sect. 3.1, part ofthis shift was caused by the ` ≈ 1800 systematic in the nominal-mission 217 × 217 spectrum used in PCP13.

The red contours in Fig. 21 show the constraints from PlanckTT+lowP. These are similar to the constraints shown in Fig. 23of PCP13, but with ns shifted to slightly higher values. The ad-dition of BAO or the Planck lensing data to Planck TT+lowPlowers the value of Ωch2, which, at fixed θ∗, increases the small-scale CMB power. To maintain the fit to the Planck temperaturepower spectrum for models with r = 0, these parameter shiftsare compensated by a change in the amplitude As and the tiltns (by about 0.4σ). The increase in ns to match the observedpower on small scales leads to a decrease in the scalar poweron large scales, allowing room for a slightly larger contribu-tion from tensor modes. The constraints shown by the blue con-tours in Fig. 21, which combine Planck lensing, BAO, and otherastrophysical data, are therefore tighter in the ns direction andshifted to slightly higher values, but marginally weaker in ther-direction. The 95 % limits on r0.002 are

r0.002 < 0.10, Planck TT+lowP, (39a)r0.002 < 0.11, Planck TT+lowP+lensing+ext, (39b)

consistent with the results reported in PCP13. Here we assumethe second-order slow-roll consistency relation for the tensorspectral index. The result in Eqs. (39a) and (39b) are mildly scaledependent, with equivalent limits on r0.05 being weaker by about5 %.

PCP13 noted a mismatch between the best-fit base ΛCDMmodel and the temperature power spectrum at multipoles ` <∼ 40,partly driven by the dip in the multipole range 20 <∼ ` <∼ 30. Ifthis mismatch is simply a statistical fluctuation of the ΛCDMmodel (and there is no compelling evidence to think otherwise),the strong Planck limit (compared to forecasts) is the result ofchance low levels of scalar mode confusion. On the other hand, ifthe dip represents a failure of the ΛCDM model, the 95 % limitsof Eqs. (39a) and (39b) may be underestimates. These issues areconsidered at greater length in Planck Collaboration XX (2016)and will not be discussed further in this paper.

As mentioned above, the Planck temperature constraints onr are model-dependent and extensions to ΛCDM can give sig-nificantly different results. For example, extra relativistic de-grees of freedom increase the small-scale damping of the CMBanisotropies at a fixed angular scale, which can be compensatedby increasing ns, allowing a larger tensor mode. This is illus-trated by the grey contours in Fig. 21, which show the constraintsfor a model with ∆Neff = 0.39. Although this value of ∆Neff isdisfavoured by the Planck data (see Sect. 6.4.1) it is not excludedat a high significance level.

This example emphasizes the need for direct tests oftensor modes based on measurements of a large-scale B-mode pattern in CMB polarization. Planck B-mode constraintsfrom the 100- and 143-GHz HFI channels, presented inPlanck Collaboration XI (2016), give a 95 % upper limit of r <∼0.27. However, at present the tightest B-mode constraints on rcome from the BKP analysis of the BICEP2/Keck field, whichcovers approximately 400, deg2 centred on RA = 0h, Dec = −57.5. These measurementsprobe the peak of the B-mode power spectrum at around ` = 100,corresponding to gravitational waves with k ≈ 0.01 Mpc−1 thatenter the horizon during recombination (i.e., somewhat smaller

0.90 0.95 1.00 1.05 1.10

ns,0.002

0.0

0.1

0.2

0.3

0.4

r 0.0

02

ΛCDM+running+tensors

ΛCDM+tensors

−0

.04

0−

0.0

24

−0

.00

80

.00

8

dn

s /d

lnk

Fig. 22. Constraints on the tensor-to-scalar ratio r0.002 inthe ΛCDM model with running, using Planck TT+lowP(samples, coloured by the running parameter), and PlanckTT+lowP+lensing+BAO (black contours). Dashed contoursshow the corresponding constraints also including the BKP B-mode likelihood. These are compared to the constraints when therunning is fixed to zero (blue contours). Parameters are plottedat k = 0.002 Mpc−1, which is approximately the scale at whichPlanck probes tensor fluctuations; however, the scalar tilt is onlyconstrained well on much smaller scales. The inflationary slow-roll consistency relation is used here for nt (though the range ofrunning allowed is much larger than would be expected in mostslow-roll models).

than the scales that contribute to the Planck temperature con-straints on r). The results of BKP give a posterior for r that peaksat r0.05 ≈ 0.05, but is consistent with r0.05 = 0. Thus, at presentthere is no convincing evidence of a primordial B-mode signal.At these low values of r, there is no longer any tension withPlanck temperature constraints.

The analysis of BKP constrains r defined relative to a fixedfiducial B-mode spectrum, and on its own does not give a use-ful constraint on either the scalar amplitude or ns. A combinedanalysis of the Planck CMB spectra and the BKP likelihood can,self-consistently, give constraints in the ns–r plane, as shown inthe right-hand panel of Fig. 21. The BKP likelihood pulls thecontours to slightly non-zero values of r, with best fits of aroundr0.002 ≈ 0.03, but at very low levels of statistical significance.The BKP likelihood also rules out the upper tail of r values al-lowed by Planck alone. The joint Planck+BKP likelihood anal-yses give the 95 % upper limits

r0.002 < 0.08, Planck TT+lowP+BKP, (40a)r0.002 < 0.09, Planck TT+lowP+lensing+ext+BKP. (40b)

The exact values of these upper limits are weakly dependent onthe details of the foreground modelling applied in the BKP anal-ysis (see BKP for further details). The results given here are forthe baseline 2-parameter model, varying the B-mode dust ampli-tude and frequency scaling, using the lowest five B-mode band-powers.

Allowing a running of the scalar spectral index as an addi-tional free parameter weakens the Planck constraints on r0.002, asshown in Fig. 22. The coloured samples in Fig. 22 illustrate how

35


a negative running allows the large-scale scalar spectral indexns,0.002 to shift towards higher values, lowering the scalar poweron large scales relative to small scales, thereby allowing a largertensor contribution. However adding the BKP likelihood, whichdirectly constrains the tensor amplitude on smaller scales, signif-icantly reduces the extent of this degeneracy leading to a 95 %upper limit of r0.002 < 0.10 even in the presence of running (i.e.,similar to the results of Eqs. 40a and 40b).

The Planck+BKP joint analysis rules out a quadratic infla-tionary potential (V(φ) ∝ m2φ2, predicting r ≈ 0.16) at over99 % confidence and reduces the allowed range of the param-eter space for models with convex potentials. Starobinsky-typemodels are an example of a wider class of inflationary theoriesin which ns − 1 = O(1/N) is not a coincidence, yet r = O(1/N2)(Roest 2014; Creminelli et al. 2015). These models have con-cave potentials, and include a variety of string-inspired modelswith exponential potentials. Models with r = O(1/N) are, how-ever, still allowed by the data, including a simple linear potentialand fractional-power monomials, as well as regions of parameterspace in between where ns − 1 = O(1/N) is just a coincidence.Models that have sub-Planckian field evolution, so satisfying theLyth bound (Lyth 1997; Garcia-Bellido et al. 2014), will typi-cally have r <∼ 2 × 10−5 for ns ≈ 0.96, and are also consistentwith the tensor constraints shown in Fig. 21. For further discus-sion of the implications of the Planck 2015 data for a wide rangeof inflationary models see Planck Collaboration XX (2016).

In summary, the Planck limits on r are consistent with theBKP limits from B-mode measurements. Both data sets are con-sistent with r = 0; however, the combined data sets yield an up-per limit to the tensor-to-scalar ratio of r ≈ 0.09 at the 95 % level.The Planck temperature constraints on r are limited by cosmicvariance. The only way of improving these limits, or poten-tially detecting gravitational waves with r <∼ 0.09, is through di-rect B-mode detection. The Planck 353-GHz polarization maps(Planck Collaboration Int. XXX 2016) show that at frequenciesof around 150 GHz, Galactic dust emission is an important con-taminant at the r ≈ 0.05 level even in the cleanest regions of thesky. BKP demonstrates further that on small regions of the skycovering a few hundred square degrees (typical of ground basedB-mode experiments), the Planck 353-GHz maps are of limiteduse as monitors of polarized Galactic dust emission because oftheir low signal-to-noise level. To achieve limits substantiallybelow r ≈ 0.05 will require observations of comparably highsensitivity over a range of frequencies, and with increased skycoverage. The forthcoming measurements from Keck Array andBICEP3 at 95 GHz and the Keck Array receivers at 220 GHzshould offer significant improvements on the current constraints.A number of other ground-based and sub-orbital experimentsshould also return high precision B-mode data within the nextfew years (see Abazajian et al. 2015a, for a review).

6.2.2. Scale dependence of primordial fluctuations

In simple single-field models of inflation, the running of thespectral index is of second order in inflationary slow-roll pa-rameters and is typically small, |dns/d ln k| ≈ (ns − 1)2 ≈ 10−3

(Kosowsky & Turner 1995). Nevertheless, it is possible to con-struct models that produce a large running over a wavenum-ber range accessible to CMB experiments, whilst simultane-ously achieving enough e-folds of inflation to solve the horizonproblem. Inflation with an oscillatory potential of sufficientlylong period, perhaps related to axion monodromy, is an exam-ple (Silverstein & Westphal 2008; Meerburg 2014; Czerny et al.2014; Minor & Kaplinghat 2015).

0.90 0.95 1.00 1.05

ns,0.002

−0.030

−0.015

0.000

0.015

dn

s/d

lnk

0.945

0.950

0.955

0.960

0.965

0.970

0.975

0.980

0.985

ns

Fig. 23. Constraints on the running of the scalar spectral in-dex in the ΛCDM model, using Planck TT+lowP (samples,coloured by the spectral index at k = 0.05 Mpc−1), and PlanckTT,TE,EE+lowP (black contours). The Planck data are consis-tent with zero running, but also allow for significant negativerunning, which gives a positive tilt on large scales and henceless power on large scales.

As reviewed in PCP13, previous CMB experiments, eitheron their own or in combination with other astrophysical data,have sometimes given hints of a non-zero running at about the2σ level (Spergel et al. 2003; Hinshaw et al. 2013; Hou et al.2014). The results of PCP13 showed a slight preference for nega-tive running at the 1.4σ level, driven almost entirely by the mis-match between the CMB temperature power spectrum at highmultipoles and the spectrum at multipoles ` <∼ 50.

The 2015 Planck results (Fig. 23) are similar to those inPCP13. Adding running as an additional parameter to baseΛCDM with r = 0, we find

dns

d ln k= −0.0084 ± 0.0082, Planck TT+lowP, (41a)

dns

d ln k= −0.0057 ± 0.0071, Planck TT,TE,EE+lowP. (41b)

There is a slight preference for negative running, which, as inPCP13, is driven by the mismatch between the high and lowmultipoles in the temperature power spectrum. However, in the2015 Planck data the tension between high and low multipolesis reduced somewhat, primarily because of changes to the HFIbeams at multipoles ` <∼ 200 (see Sect. 3.1). A consequenceof this reduced tension can be seen in the 2015 constraints onmodels that include tensor fluctuations in addition to running:

dns

d ln k= −0.0126+0.0098

−0.0087, Planck TT+lowP; (42a)

dns

d ln k= −0.0085 ± 0.0076, Planck TT,TE,EE+lowP; (42b)

dns

d ln k= −0.0065 ± 0.0076, Planck TT+lowP+lensing

+ext+BKP. (42c)

PCP13 found an approximately 2σ pull towards negative run-ning for these models. This tension is reduced to about 1σ withthe 2015 Planck data, and to lower values when we include theBKP likelihood, which reduces the range of allowed tensor am-plitudes.

36


In summary, the Planck data are consistent with zero runningof the scalar spectral index. However, as illustrated in Fig. 23,the Planck data still allow running at roughly the 10−2 level,i.e., an order of magnitude higher than expected in simple in-flationary models. One way of potentially improving these con-straints is to extend the wavenumber range from CMB scalesto smaller scales using additional astrophysical data, for exam-ple by using measurements of the Lyα flux power spectrumof high-redshift quasars (as in the first year WMAP analysis,Spergel et al. 2003). Palanque-Delabrouille et al. (2015) have re-cently reported an analysis of a large sample of quasar spec-tra from the SDSSIII/BOSS survey. These authors find a lowvalue of the scalar spectral index ns = 0.928 ± 0.012 (stat.) ±(0.02) (syst.) on scales of k ≈ 1 Mpc−1. To extract physical pa-rameters, the Lyα power spectra need to be calibrated againstnumerical hydrodynamical simulations. The large systematic er-ror in this spectral index determination is dominated by the fi-delity of the hydrodynamic simulations and by the splicing usedto achieve high resolution over large scales. These uncertain-ties need to be reduced before addressing the consistency of Lyαresults with CMB measurements of the running of the spectralindex.

6.2.3. Isocurvature perturbations

A key prediction of single-field inflation is that the primordialperturbations are adiabatic. More generally, the observed fluc-tuations will be adiabatic in any model in which the curvatureperturbations were the only super-horizon perturbations left bythe time that dark matter (and other matter) first decoupled, orwas produced by decay. The different matter components thenall have perturbations proportional to the curvature perturbation,so there are no isocurvature perturbations. However, it is possibleto produce an observable amount of isocurvature modes by hav-ing additional degrees of freedom present during inflation andthrough reheating. For example, the curvaton model can gener-ate correlated adiabatic and isocurvature modes from a secondfield (Mollerach 1990; Lyth & Wands 2002).

Isocurvature modes describe relative perturbations betweenthe different species (Bucher et al. 2001b), with perhaps the sim-plest being a perturbation in the baryonic or dark matter sector(relative to the radiation). However, only one total matter isocur-vature mode is observable in the linear CMB (in the accurateapproximation in which the baryons are pressureless); a com-pensated mode (between the baryons and the cold dark matter)with δρb = −δρc has no net density perturbation, and producesno CMB anisotropies (Gordon & Lewis 2003; Grin et al. 2011;Grin et al. 2014). It is possible to generate isocurvature modes inthe neutrino sector; however, this requires interaction of an ad-ditional perturbed super-horizon field with neutrinos after theyhave decoupled, and hence is harder to achieve. Finally, neutrinovelocity potential and vorticity modes are other possible con-sistent perturbations to the photon-neutrino fluid after neutrinodecoupling. However, they are essentially impossible to excite,since they consist of photon and neutrino fluids coherently mov-ing in opposite directions on super-horizon scales (although therelative velocity would have been zero before neutrino decou-pling).

Planck Collaboration XXII (2014) presented constraints ona variety of general isocurvature models using the Planck tem-perature data, finding consistency with adiabaticity, though withsome mild preference for isocurvature models that reduce thepower at low multipoles to provide a better match to the Plancktemperature spectrum at ` <∼ 50. For matter isocurvature pertur-

0.945 0.960 0.975 0.990

ns

−0.012

−0.006

0.000

0.006

α

Planck TT+lowP


Fig. 24. Constraints on the correlated matter isocurvature modeamplitude parameter α, where α = 0 corresponds to purely adia-batic perturbations. The Planck temperature data slightly favournegative values, since this lowers the large-scale anisotropies;however, the polarization signal from an isocurvature mode isdistinctive and the Planck polarization data significantly shrinkthe allowed region around the value α = 0 corresponding to adi-abatic perturbations.

bations, the photons are initially unperturbed but perturbationsdevelop as the Universe becomes more matter dominated. As aresult, the phase of the acoustic oscillations differs from adia-batic modes; this is most clearly distinctive with the addition ofpolarization data (Bucher et al. 2001a)

An extended analysis of isocurvature models is given inPlanck Collaboration XX (2016). Here we focus on a simple il-lustrative case of a totally-correlated matter isocurvature mode.We define an isocurvature amplitude parameter α, such that27

S m = sgn(α)

√|α|

1 − |α|ζ, (43)

where ζ is the primordial curvature perturbation. Here S m is thetotal matter isocurvature mode, defined as the observable sumof the baryon and CDM isocurvature modes, i.e., S m = S c +S b(ρb/ρc), where

S i ≡δρi

ρi−

3δργ4ργ

. (44)

All modes are assumed to have a power spectrum with thesame spectral index ns, so that α is independent of scale. Forpositive α this agrees with the definitions in Bean et al. (2006)and Larson et al. (2011) for α−1, but also allows for the corre-lation to have the opposite sign. Approximately, sgn(α)α2 ≈

Bc, where Bc is the CDM version of the amplitude defined asin Amendola et al. (2002). Note that in our conventions, nega-tive values of α lower the Sachs–Wolfe contribution to the large-scale TT power spectrum. We caution the reader that this con-vention differs from some others, e.g., Larson et al. (2011).

27Planck Collaboration XX (2016) gives equivalent one-tailed con-straints on βiso = |α|, where the correlated and anti-correlated cases areconsidered separately.

37


10 20 30 40 50 60

`

0

500

1000

1500

2000

DT

T`

[µK

2]

-0.0

10

-0.0

05

0.0

00

α

50 100 150 200

`

0.0

0.2

0.4

0.6

0.8

1.0

1.2

DE

E`

[µK

2]

-0.0

10

-0.0

05

0.0

00

α

Fig. 25. Power spectra drawn from the Planck TT+lowP posterior for the correlated matter isocurvature model, colour-coded by thevalue of the isocurvature amplitude parameter α, compared to the Planck data points. The left-hand figure shows how the negatively-correlated modes lower the large-scale temperature spectrum, slightly improving the fit at low multipoles. Including polarization,the negatively-correlated modes are disfavoured, as illustrated at the first acoustic peak in EE on the right-hand plot. Data points at` < 30 are not shown for polarization, as they are included with both the default temperature (i.e., TT+lowP) and polarization (i.e.,TT,TE,EE+lowP) likelihood combinations.

Planck constraints on the correlated isocurvature amplitudeare shown in Fig. 24, with and without high-multipole polariza-tion. The corresponding marginalized limit from the temperaturedata is

α = −0.0025+0.0035−0.0047 (95%,Planck TT+lowP), (45)

which is significantly tightened around zero when Planck polar-ization information is included at high multipoles:

α = 0.0003+0.0016−0.0012 (95%,Planck TT,TE,EE+lowP). (46)

This strongly limits the isocurvature contribution to be less thanabout 3 % of the adiabatic modes. Figure 25 shows how modelswith negative correlation parameter, α, fit the temperature data atlow multipoles slightly better than models with α = 0; however,these models are disfavoured from the corresponding change inthe polarization acoustic peaks.

In this model most of the gain in sensitivity comes fromrelatively large scales, ` <∼ 300, where the correlated isocur-vature modes with delayed phase change the first polarizationacoustic peak (` ≈ 140) significantly more than in tempera-ture (Bucher et al. 2001a). The polarization data are not entirelyrobust to systematics on these scales, but in this case the resultappears to be quite stable between the different likelihood codes.However, it should be noted that a particularly low point in theT E spectrum at ` ≈ 160 (see Fig. 3) pulls in the direction ofpositive α, and could be giving an artificially strong constraint ifthis were caused by an unidentified systematic.

6.2.4. Curvature

The simplifying assumptions of large-scale homogeneity andisotropy lead to the familiar Friedman-Lemaıtre-Robertson-Walker (FLRW) metric that appears to be an accurate descriptionof our Universe. The base ΛCDM cosmology assumes an FLRWmetric with a flat 3-space. This is a very restrictive assumptionthat needs to be tested empirically. In this subsection, we investi-gate constraints on the parameter ΩK , where for ΛCDM models

0.30 0.45 0.60 0.75

Ωm

0.30

0.45

0.60

0.75

ΩΛ

+TE+EE

+lensing

+lensing+BAO

40

44

48

52

56

60

64

68

H0

Fig. 26. Constraints in the Ωm–ΩΛ plane from the PlanckTT+lowP data (samples; colour-coded by the value of H0) andPlanck TT,TE,EE+lowP (solid contours). The geometric degen-eracy between Ωm and ΩΛ is partially broken because of the ef-fect of lensing on the temperature and polarization power spec-tra. These limits are improved significantly by the inclusionof the Planck lensing reconstruction (blue contours) and BAO(solid red contours). The red contours tightly constrain the ge-ometry of our Universe to be nearly flat.

ΩK ≡ 1 − Ωm − ΩΛ. For FLRW models ΩK > 0 correspondsto negatively-curved 3-geometries while ΩK < 0 correspondsto positively-curved 3-geometries. Even with perfect data withinour past lightcone, our inference of the curvature ΩK is limitedby the cosmic variance of curvature perturbations that are stillsuper-horizon at the present, since these cannot be distinguishedfrom background curvature within our observable volume.

38


The parameter ΩK decreases exponentially with time duringinflation, but grows only as a power law during the radiationand matter-dominated phases, so the standard inflationary pre-diction has been that curvature should be unobservably smalltoday. Nevertheless, by fine-tuning parameters it is possible todevise inflationary models that generate open (e.g., Bucher et al.1995; Linde 1999) or closed universes (e.g., Linde 2003). Evenmore speculatively, there has been interest recently in multi-verse models, in which topologically-open “pocket universes”form by bubble nucleation (e.g., Coleman & De Luccia 1980;Gott 1982) between different vacua of a “string landscape” (e.g.,Freivogel et al. 2006; Bousso et al. 2015). Clearly, the detectionof a significant deviation from ΩK = 0 would have profoundconsequences for inflation theory and fundamental physics.

The Planck power spectra give the constraint

ΩK = −0.052+0.049−0.055 (95%,Planck TT+lowP). (47)

The well-known geometric degeneracy (Bond et al. 1997;Zaldarriaga et al. 1997) allows for the small-scale linear CMBspectrum to remain almost unchanged if changes in ΩK are com-pensated by changes in H0 to obtain the same angular diam-eter distance to last scattering. The Planck constraint is there-fore mainly determined by the (wide) priors on H0, and the ef-fect of lensing smoothing on the power spectra. As discussed inSect. 5.1, the Planck temperature power spectra show a slightpreference for more lensing than expected in the base ΛCDMcosmology, and since positive curvature increases the amplitudeof the lensing signal, this preference also drives ΩK towards neg-ative values.

Taken at face value, Eq. (47) represents a detection of posi-tive curvature at just over 2σ, largely via the impact of lensingon the power spectra. One might wonder whether this is mainlya parameter volume effect, but that is not the case, since the bestfit closed model has ∆χ2 ≈ 6 relative to base ΛCDM, and the fitis improved over almost all the posterior volume, with the meanimprovement being 〈∆χ2〉 ≈ 5 (very similar to the phenomeno-logical case of ΛCDM+AL). Addition of the Planck polarizationspectra shifts ΩK towards zero by ∆ΩK ≈ 0.015:

ΩK = −0.040+0.038−0.041 (95%,Planck TT,TE,EE+lowP), (48)

but ΩK remains negative at just over 2σ.What’s more, the lensing reconstruction from Planck mea-

sures the lensing amplitude directly and, as discussed inSect. 5.1, this does not prefer more lensing than base ΛCDM.The combined constraint shows impressive consistency with aflat universe:

ΩK = −0.005+0.016−0.017 (95%,Planck TT+lowP+lensing). (49)

The dramatic improvement in the error bar is another illustrationof the power of the lensing reconstruction from Planck.

The constraint can be sharpened further by adding externaldata that break the main geometric degeneracy. Combining thePlanck data with BAO, we find

ΩK = 0.000 ± 0.005 (95%, Planck TT+lowP+lensing+BAO).(50)

This constraint is unchanged at the quoted precision if we addthe JLA supernovae data and the H0 prior of Eq. (30).

Figure 26 illustrates these results in the Ωm–ΩΛ plane. Weadopt Eq. (50) as our most reliable constraint on spatial curva-ture. Our Universe appears to be spatially flat to a 1σ accuracyof 0.25 %.

−1.2 −1.0 −0.8 −0.6

w0

−1.6

−0.8

0.0

0.8

wa

65.6

66.4

67.2

68.0

68.8

69.6

70.4

71.2

H0

Fig. 27. Samples from the distribution of the dark energy pa-rameters w0 and wa using Planck TT+lowP+BAO+JLA data,colour-coded by the value of the Hubble parameter H0. Contoursshow the corresponding 68 % and 95 % limits. Dashed grey linesintersect at the point in parameter space corresponding to a cos-mological constant.

6.3. Dark energy

The physical explanation for the observed accelerated expansionof the Universe is currently not known. In standard ΛCDM theacceleration is provided by a cosmological constant, i.e., an ad-ditional fluid satisfying an equation of state w ≡ pDE/ρDE = −1.However, there are many possible alternatives, typically de-scribed either in terms of extra degrees of freedom associatedwith scalar fields or modifications of general relativity on cos-mological scales (for reviews see, e.g., Copeland et al. 2006;Tsujikawa 2010). A detailed study of these models and the con-straints imposed by Planck and other data are presented in a sep-arate paper, Planck Collaboration XIV (2016).

Here we will limit ourselves to the most basic extensionsof ΛCDM, which can be phenomenologically described interms of the equation of state parameter w alone. Specificallywe will use the camb implementation of the “parameterizedpost-Friedmann” (PPF) framework of Hu & Sawicki (2007) andFang et al. (2008) to test whether there is any evidence that wvaries with time. This framework aims to recover the behaviourof canonical (i.e., those with a standard kinetic term) scalar fieldcosmologies minimally coupled to gravity when w ≥ −1, andaccurately approximates them for values w ≈ −1. In these mod-els the speed of sound is equal to the speed of light, so that theclustering of the dark energy inside the horizon is strongly sup-pressed. The advantage of using the PPF formalism is that it ispossible to study the phantom domain, w < −1, including transi-tions across the “phantom barrier,” w = −1, which is not possiblefor canonical scalar fields.

The CMB temperature data alone do not tightly constrain w,because of a strong geometrical degeneracy, even for spatially-flat models. From Planck we find

w = −1.54+0.62−0.50 (95%,Planck TT+lowP), (51)

i.e., almost a 2σ shift into the phantom domain. This is partly,but not entirely, a parameter volume effect, with the average ef-fective χ2 improving by 〈∆χ2〉 ≈ 2 compared to base ΛCDM.

39


This is consistent with the preference for a higher lensing am-plitude discussed in Sect. 5.1.2, improving the fit in the w < −1region, where the lensing smoothing amplitude becomes slightlylarger. However, the lower limit in Eq. (51) is largely determinedby the (arbitrary) prior H0 < 100 km s−1Mpc−1, chosen for theHubble parameter. Much of the posterior volume in the phan-tom region is associated with extreme values for cosmologicalparameters, which are excluded by other astrophysical data. Themild tension with base ΛCDM disappears as we add more datathat break the geometrical degeneracy. Adding Planck lensingand BAO, JLA and H0 (“ext”) gives the 95 % constraints

w = −1.023+0.091−0.096 Planck TT+lowP+ext, (52a)

w = −1.006+0.085−0.091 Planck TT+lowP+lensing+ext, (52b)

w = −1.019+0.075−0.080 Planck TT,TE,EE+lowP+lensing+ext.

(52c)

The addition of Planck lensing, or using the full Planck tem-perature+polarization likelihood together with the BAO, JLA,and H0 data does not substantially improve the constraint ofEq. (52a). All of these data set combinations are compatible withthe base ΛCDM value of w = −1. In PCP13, we conservativelyquoted w = −1.13+0.24

−0.25, based on combining Planck with BAO,as our most reliable limit on w. The errors in Eqs. (52a)–(52c) aresubstantially smaller, mainly because of the addition of the JLASNe data, which offer a sensitive probe of the dark energy equa-tion of state at z <∼ 1. In PCP13, the addition of the SNLS SNedata pulled w into the phantom domain at the 2σ level, reflectingthe tension between the SNLS sample and the Planck 2013 baseΛCDM parameters. As noted in Sect. 5.3, this discrepancy is nolonger present, following improved photometric calibrations ofthe SNe data in the JLA sample. One consequence of this is thetightening of the errors in Eqs. (52a)–(52c) around the ΛCDMvalue w = −1 when we combine the JLA sample with Planck.

If w differs from −1, it is likely to change with time. Weconsider here the case of a Taylor expansion of w at first order inthe scale factor, parameterized by

w = w0 + (1 − a)wa. (53)

More complex models of dynamical dark energy are discussedin Planck Collaboration XIV (2016). Figure 27 shows the 2Dmarginalized posterior distribution for w0 and wa for the com-bination Planck+BAO+JLA. The JLA SNe data are again cru-cial in breaking the geometrical degeneracy at low redshift andwith these data we find no evidence for a departure from thebase ΛCDM cosmology. The points in Fig. 27 show samplesfrom these chains colour-coded by the value of H0. From theseMCMC chains, we find H0 = (68.2 ± 1.1) km s−1Mpc−1. Muchhigher values of H0 would favour the phantom regime, w < −1.

As pointed out in Sects. 5.5.2 and 5.6 the CFHTLenS weaklensing data are in tension with the Planck base ΛCDM param-eters. Examples of this tension can be seen in investigations ofdark energy and modified gravity, since some of these modelscan modify the growth rate of fluctuations from the base ΛCDMpredictions. This tension can be seen even in the simple modelof Eq. (53). The green regions in Fig. 28 show 68 % and 95 %contours in the w0–wa plane for Planck TT+lowP combined withthe CFHTLenS H13 data. In this example, we have applied ultra-conservative cuts, excluding ξ− entirely and excluding measure-ments with θ < 17′ in ξ+ for all tomographic redshift bins. Asdiscussed in Planck Collaboration XIV (2016), with these cutsthe CFHTLenS data are insensitive to modelling the nonlinearevolution of the power spectrum, but this reduction in sensitiv-ity comes at the expense of reducing the statistical power of the

−2 −1 0 1

w0

−3

−2

−1

0

1

2

wa

Planck TT+lowP+ext

Planck TT+lowP+WLPlanck TT+lowP+WL+H0

Fig. 28. Marginalized posterior distributions for (w0,wa) for var-ious data combinations. We show Planck TT+lowP in combi-nation with BAO, JLA, H0 (“ext”), and two data combinationsthat add the CFHTLenS data with ultra-conservative cuts as de-scribed in the text (denoted “WL”). Dashed grey lines show theparameter values corresponding to a cosmological constant.

weak lensing data. Nevertheless, Fig. 28 shows that the combina-tion of Planck+CFHTLenS pulls the contours into the phantomdomain and is discrepant with base ΛCDM at about the 2σ level.The Planck+CFHTLenS data also favour a high value of H0. Ifwe add the (relatively weak) H0 prior of Eq. (30), the contours(shown in cyan) in Fig. 28 shift towards w = −1. It thereforeseems unlikely that the tension between Planck and CFHTLenScan be resolved by allowing a time-variable equation of state fordark energy.

A much more extensive investigation of models of darkenergy and also models of modified gravity can be found inPlanck Collaboration XIV (2016). The main conclusions of thatanalysis are as follows:

• an investigation of more general time-variations of the equa-tion of state shows a high degree of consistency with w = −1;

• a study of several dark energy and modified gravity modelseither finds compatibility with base ΛCDM, or mild tensions,which are driven mainly by external data sets.

6.4. Neutrino physics and constraints on relativisticcomponents

In the following subsections, we update Planck constraints onthe mass of standard (active) neutrinos, additional relativistic de-grees of freedom, models with a combination of the two, andmodels with massive sterile neutrinos. In each subsection weemphasize the Planck-only constraint, and the implications ofthe Planck result for late-time cosmological parameters mea-sured from other observations. We then give a brief discussion oftensions between Planck and some discordant external data, andassess whether any of these model extensions can help to resolvethem. Finally we provide constraints on neutrino interactions.

40


6.4.1. Constraints on the total mass of active neutrinos

Detection of neutrino oscillations has proved that neutri-nos have mass (see, e.g., Lesgourgues & Pastor 2006 andNakamura & Petcov 2014 for reviews). The Planck base ΛCDMmodel assumes a normal mass hierarchy with

∑mν ≈ 0.06 eV

(dominated by the heaviest neutrino mass eigenstate) but thereare other possibilities, including a degenerate hierarchy with∑

mν >∼ 0.1 eV. At this time there are no compelling theoreti-cal reasons to strongly prefer any of these possibilities, so allow-ing for larger neutrino masses is perhaps one of the most well-motivated extensions to base ΛCDM considered in this paper.There has also been significant interest recently in larger neu-trino masses as a possible way to lower σ8 (the late-time fluctua-tion amplitude), and thereby reconcile Planck with weak lensingmeasurements and the abundance of rich clusters (see Sects. 5.5and 5.6). Though model dependent, neutrino mass constraintsfrom cosmology are already significantly stronger than thosefrom tritium β-decay experiments (see, e.g., Drexlin et al. 2013).

Here we give constraints assuming three species of degener-ate massive neutrinos, neglecting the small differences in massexpected from the observed mass splittings. At the level of sensi-tivity of Planck this is an accurate approximation, but note that itdoes not quite match continuously on to the base ΛCDM model(which assumes two massless and one massive neutrino with∑

mν = 0.06 eV). We assume that the neutrino mass is con-stant, and that the distribution function is Fermi-Dirac with zerochemical potential.

Masses well below 1 eV have only a mild effect on the shapeof the CMB power spectra, since they became non-relativistic af-ter recombination. The effect on the background cosmology canbe compensated by changes in H0, to ensure the same observedacoustic peak scale θ∗. There is, however, some sensitivity ofthe CMB anisotropies to neutrino masses as the neutrinos startto become less relativistic at recombination (modifying the earlyISW effect), and from the late-time effect of lensing on the powerspectrum. The Planck power spectrum (95 %) constraints are∑

mν < 0.72 eV Planck TT+lowP, (54a)∑mν < 0.21 eV Planck TT+lowP+BAO, (54b)∑mν < 0.49 eV Planck TT,TE,EE+lowP, (54c)∑mν < 0.17 eV Planck TT,TE,EE+lowP+BAO. (54d)

The Planck TT+lowP constraint has a broad tail to high masses,as shown in Fig. 29, which also illustrates the acoustic scaledegeneracy with H0. Larger masses imply a lower σ8 throughthe effects of neutrino free-streaming on structure formation,but the larger masses also require a lower Hubble constant,leading to possible tensions with direct measurements of H0.Masses below about 0.4 eV can provide an acceptable fit tothe direct H0 measurements, and adding the BAO data helpsto break the acoustic scale degeneracy and tightens the con-straint on

∑mν substantially. Adding Planck polarization data at

high multipoles produces a relatively small improvement to thePlanck TT+lowP+BAO constraint (and the improvement is evensmaller with the alternative CamSpec likelihood), so we considerthe TT results to be our most reliable constraints.

The constraint of Eq. (54b) is consistent with the 95 % limitof

∑mν < 0.23 eV reported in PCP13 for Planck+BAO. The

limits are similar because the linear CMB is insensitive to themass of neutrinos that are relativistic at recombination. There islittle to be gained from improved measurement of the CMB tem-perature power spectra, though improved external data can help

0.0 0.4 0.8 1.2 1.6

Σmν [eV]

55

60

65

70

75

H0

[km

s−1

Mp

c−1]

0.60

0.64

0.68

0.72

0.76

0.80

0.84

σ8

Fig. 29. Samples from the Planck TT+lowP posterior in the∑mν–H0 plane, colour-coded by σ8. Higher

∑mν damps the

matter fluctuation amplitude σ8, but also decreases H0. Thegrey bands show the direct measurement, H0 = (70.6 ±3.3) km s−1Mpc−1, Eq. (30). Solid black contours show theconstraint from Planck TT+lowP+lensing (which mildly preferslarger masses), and filled contours show the constraints fromPlanck TT+lowP+lensing+BAO.

to break the geometric degeneracy to higher precision. CMBlensing can also provide additional information at lower red-shifts, and future high-resolution CMB polarization measure-ments that accurately reconstruct the lensing potential can probemuch smaller masses (see, e.g. Abazajian et al. 2015b).

As discussed in detail in PCP13 and Sect. 5.1, the PlanckCMB power spectra prefer somewhat more lensing smoothingthan predicted in ΛCDM (allowing the lensing amplitude to varygives AL > 1 at just over 2σ). The neutrino mass constraintfrom the power spectra is therefore quite tight, since increas-ing the neutrino mass lowers the predicted smoothing even fur-ther compared to base ΛCDM. On the other hand the lensingreconstruction data, which directly probes the lensing power,prefers lensing amplitudes slightly below (but consistent with)the base ΛCDM prediction (Eq. 18). The Planck+lensing con-straint therefore pulls the constraints slightly away from zero to-wards higher neutrino masses, as shown in Fig. 30. Although theposterior has less weight at zero, the lensing data are incompati-ble with very large neutrino masses so the Planck+lensing 95 %limit is actually tighter than the Planck TT+lowP result:∑

mν < 0.68 eV (95%,Planck TT+lowP+lensing). (55)

Adding the polarization spectra improves this constraint slightlyto∑

mν < 0.59 eV (95%,Planck TT,TE,EE+lowP+lensing).(56)

We take the combined constraint that further includes BAO,JLA, and H0 (“ext”) as our best limit:∑

mν < 0.23 eV

Ωνh2 < 0.0025

95%, Planck TT+lowP+lensing+ext.

(57)

41


0.00 0.25 0.50 0.75 1.00

Σmν [eV]

0

1

2

3

4

5

6

7

8

Pro

bab

ility

den

sity

[eV−

1]

Planck TT+lowP

+lensing

+ext


+lensing

+ext

Fig. 30. Constraints on the sum of the neutrino masses for vari-ous data combinations.

This is slightly weaker than the constraint from PlanckTT,TE,EE+lowP+lensing+BAO (which is tighter in both theCamSpec and Plik likelihoods), but is immune to low level sys-tematics that might affect the constraints from the Planck polar-ization spectra. Equation (57) is therefore a conservative limit.Marginalizing over the range of neutrino masses, the Planck con-straints on the late-time parameters are28

H0 = 67.7 ± 0.6

σ8 = 0.810+0.015−0.012

Planck TT+lowP+lensing+ext. (58)

For this restricted range of neutrino masses, the impact on theother cosmological parameters is small and, in particular, lowvalues of σ8 will remain in tension with the parameter spacepreferred by Planck.

The constraint of Eq. (57) is weaker than the constraint ofEq. (54b) excluding lensing, but there is no good reason to disre-gard the Planck lensing information while retaining other astro-physical data. The CMB lensing signal probes very-nearly lin-ear scales and passes many consistency checks over the multi-pole range used in the Planck lensing likelihood (see Sect. 5.1and Planck Collaboration XV 2016). The situation with galaxyweak lensing is rather different, as discussed in Sect. 5.5.2. Inaddition to possible observational systematics, the weak lensingdata probe lower redshifts than CMB lensing, and smaller spa-tial scales, where uncertainties in modelling nonlinearities in thematter power spectrum and baryonic feedback become impor-tant (Harnois-Deraps et al. 2015).

A larger range of neutrino masses was found by Beutler et al.(2014) using a combination of RSD, BAO, and weak lens-ing information. The tension between the RSD results andbase ΛCDM was subsequently reduced following the analysisof Samushia et al. (2014), as shown in Fig. 17. Galaxy weaklensing and some cluster constraints remain in tension with baseΛCDM, and we discuss possible neutrino resolutions of theseproblems in Sect. 6.4.4.

28To simplify the displayed equations, H0 is given in units ofkm s−1Mpc−1 in this section.

2.0 2.5 3.0 3.5 4.0 4.5

Neff

60

66

72

78

H0

[km

s−1

Mp

c−1]

0.780

0.795

0.810

0.825

0.840

0.855

0.870

0.885

0.900

σ8

Fig. 31. Samples from Planck TT+lowP chains in the Neff–H0plane, colour-coded by σ8. The grey bands show the constraintH0 = (70.6 ± 3.3) km s−1Mpc−1 of Eq. (30). Notice that higherNeff brings H0 into better consistency with direct measurements,but increases σ8. Solid black contours show the constraints fromPlanck TT,TE,EE+lowP+BAO. Models with Neff < 3.046 (leftof the solid vertical line) require photon heating after neutrinodecoupling or incomplete thermalization. Dashed vertical linescorrespond to specific fully-thermalized particle models, for ex-ample one additional massless boson that decoupled around thesame time as the neutrinos (∆Neff ≈ 0.57), or before muonannihilation (∆Neff ≈ 0.39), or an additional sterile neutrinothat decoupled around the same time as the active neutrinos(∆Neff ≈ 1).

Another way of potentially improving neutrino mass con-straints is to use measurements of the Lyα flux power spectrumof high-redshift quasars. Palanque-Delabrouille et al. (2015)have recently reported an analysis of a large sample of quasarspectra from the SDSSIII/BOSS survey. When combining theirresults with 2013 Planck data, these authors find a bound

∑mν <

0.15 eV (95 % CL), compatible with the results presented in thissection.

An exciting future prospect is the possible direct detectionof non-relativistic cosmic neutrinos by capture on tritium, forexample with the PTOLEMY experiment (Cocco et al. 2007;Betts et al. 2013; Long et al. 2014). Unfortunately, for the massrange

∑mν < 0.23 eV preferred by Planck, detection with the

first generation experiment will be extremely difficult.

6.4.2. Constraints on Neff

Dark radiation density in the early Universe is usually parame-terized by Neff , defined so that the total relativistic energy densityin neutrinos and any other dark radiation is given in terms of thephoton density ργ at T 1 MeV by

ρ = Neff

78

(4

11

)4/3

ργ. (59)

The numerical factors in this equation are included so thatNeff = 3 for three standard model neutrinos that were thermal-ized in the early Universe and decoupled well before electron-positron annihilation. The standard cosmological prediction is

42


actually Neff = 3.046, since neutrinos are not completely de-coupled at electron-positron annihilation and are subsequentlyslightly heated (Mangano et al. 2002).

In this section we focus on additional energy density frommassless particles. In addition to massless sterile neutrinos, a va-riety of other particles could contribute to Neff . We assume thatthe additional massless particles are produced well before re-combination, and neither interact nor decay, so that their energydensity scales with the expansion exactly like massless neutri-nos. An additional ∆Neff = 1 could correspond to a fully ther-malized sterile neutrino that decoupled at T <∼ 100 MeV; for ex-ample, any sterile neutrino with mixing angles large enough toprovide a potential resolution to short-baseline reactor neutrinooscillation anomalies would most likely thermalize rapidly in theearly Universe. However, this solution to the neutrino oscillationanomalies requires approximately 1-eV sterile neutrinos, ratherthan the massless case considered in this section; exploration ofthe two parameters Neff and

∑mν is reported in Sect. 6.4.3. For

a review of sterile neutrinos see Abazajian et al. (2012).More generally the additional radiation does not need to be

fully thermalized, for example there are many possible models ofnon-thermal radiation production via particle decays (see, e.g.,Hasenkamp & Kersten 2013; Conlon & Marsh 2013). The radi-ation could also be produced at temperatures T > 100 MeV,in which case typically ∆Neff < 1 for each additional species,since heating by photon production at muon annihilation (cor-responding to T ≈ 100 MeV) decreases the fractional impor-tance of the additional component at the later times relevant forthe CMB. For particles produced at T 100 MeV the den-sity would be diluted even more by numerous phase transitionsand particle annihilations, and give ∆Neff 1. Furthermore,if the particle is not fermionic, the factors entering the entropyconservation equation are different, and even thermalized par-ticles could give specific fractional values of ∆Neff . For exam-ple Weinberg (2013) considers the case of a thermalized mass-less boson, which contributes ∆Neff = 4/7 ≈ 0.57 if it decouplesin the range 0.5 MeV < T < 100 MeV like the neutrinos, or∆Neff ≈ 0.39 if it decouples at T > 100 MeV (before the photonproduction at muon annihilation, hence undergoing fractional di-lution).

In this paper we follow the usual phenomenological ap-proach, where one constrains Neff as a free parameter with awide flat prior, although we comment on a few discrete casesseparately below. Values of Neff < 3.046 are less well motivated,since they would require the standard neutrinos to be incom-pletely thermalized or additional photon production after neu-trino decoupling, but we include this range for completeness.

Figure 31 shows that Planck is entirely consistent with thestandard value Neff = 3.046. However, a significant density ofadditional radiation is still allowed, with the (68 %) constraints

Neff = 3.13 ± 0.32 Planck TT+lowP, (60a)Neff = 3.15 ± 0.23 Planck TT+lowP+BAO, (60b)Neff = 2.99 ± 0.20 Planck TT,TE,EE+lowP, (60c)Neff = 3.04 ± 0.18 Planck TT,TE,EE+lowP+BAO. (60d)

Notice the significantly tighter constraint with the inclusion ofPlanck high-` polarization, with ∆Neff < 1 at over 4σ fromPlanck alone. This constraint is not very stable between like-lihoods, with the CamSpec likelihood giving a roughly 0.8σlower value of Neff . However, the strong limit from polarizationis also consistent with the joint Planck TT+lowP+BAO result,so Eq. (60b) leads to the robust conclusion that ∆Neff < 1 at over

3σ. The addition of Planck lensing has very little effect on thisconstraint.

For Neff > 3, the Planck data favour higher values of theHubble parameter than the Planck base ΛCDM value, which asdiscussed in Sect. 5.4 may be in better agreement with somedirect measurements of H0 . This is because Planck accuratelymeasures the acoustic scale r∗/DA; increasing Neff means (viathe Friedmann equation) that the early Universe expands faster,so the sound horizon at recombination, r∗, is smaller and hencerecombination has to be closer (larger H0 and hence smallerDA) for it to subtend the same angular size observed by Planck.However, models with Neff > 3 and a higher Hubble constantalso have higher values of the fluctuation amplitudeσ8, as shownby the coloured samples in Fig. 31. As a result, these models in-crease the tensions between the CMB measurements and astro-physical measurements of σ8 discussed in Sect. 5.6. It thereforeseems unlikely that additional radiation alone can help to resolvetensions with large-scale structure data.

The energy density in the early Universe can also be probedby the predictions of BBN. In particular ∆Neff > 0 increasesthe primordial expansion rate, leading to earlier freeze-out, witha higher neutron density and hence a greater abundance of he-lium and deuterium after BBN has completed. A detailed discus-sion of the implications of Planck for BBN is given in Sect. 6.5.Observations of both the primordial helium and deuterium abun-dance are compatible with the predictions of standard BBNfor the Planck base ΛCDM value of the baryon density. ThePlanck+BBN constraints on Neff (Eqs. 75 and 76) are compati-ble, and slightly tighter than Eq. (60b).

Although there is a large continuous range of plausible Neff

values, it is worth mentioning briefly a few of the discrete valuesfrom fully thermalized models. This serves as an indication ofhow strongly Planck prefers base ΛCDM, and also how the in-ferred values of other cosmological parameters might be affectedby this particular extension to base ΛCDM. As discussed above,one fully thermalized neutrino (∆Neff ≈ 1) is ruled out at over3σ, and is disfavoured by ∆χ2 ≈ 8 compared to base ΛCDM byPlanck TT+lowP, and much more strongly in combination withPlanck high-` polarization or BAO data. The thermalized bosonmodels that give ∆Neff = 0.39 or ∆Neff = 0.57 are disfavouredby ∆χ2 ≈ 1.5 and ∆χ2 ≈ 3, respectively, and are therefore notstrongly excluded. We focus on the former, since it is also consis-tent with the Planck TT+lowP+BAO constraint at 2σ. As shownin Fig. 31, larger Neff corresponds to a region of parameter spacewith significantly higher Hubble parameter,

H0 = 70.6±1.0 (68%,Planck TT+lowP; ∆Neff = 0.39). (61)

This can be compared to the direct measurements of H0 dis-cussed in Sect. 5.4. Evidently, Eq. (61) is consistent with theH0 prior adopted in this paper (Eq. 30), but this example showsthat an accurate direct measurement of H0 can potentially pro-vide evidence for new physics beyond that probed by Planck. Asshown in Fig. 31, the ∆Neff = 0.39 cosmology also has a signif-icantly higher small-scale fluctuation amplitude and the spectralindex ns is also bluer, with

σ8 = 0.850 ± 0.015ns = 0.983 ± 0.006

Planck TT+lowP; ∆Neff = 0.39. (62)

Theσ8 range in this model is higher than preferred by the Plancklensing likelihood in base ΛCDM. However, the fit to the Plancklensing likelihood is model dependent and the lensing degener-acy direction also associates high H0 and low Ωm values withhigher σ8. The joint Planck TT+lowP+lensing constraint does

43


0.0 0.4 0.8 1.2 1.6

meffν, sterile [eV]

3.3

3.6

3.9

4.2

Neff

0.5

1.0

2.0

5.0

0.66

0.69

0.72

0.75

0.78

0.81

0.84

0.87

0.90

σ8

Fig. 32. Samples from Planck TT+lowP in the Neff–meffν, sterile

plane, colour-coded by σ8, for models with one massive ster-ile neutrino family, with effective mass meff

ν, sterile, and the threeactive neutrinos as in the base ΛCDM model. The physical massof the sterile neutrino in the thermal scenario, mthermal

sterile , is con-stant along the grey dashed lines, with the indicated mass ineV; the grey shading shows the region excluded by our priormthermal

sterile < 10 eV, which cuts out most of the area where the neu-trinos behave nearly like dark matter. The physical mass in theDodelson-Widrow scenario, mDW

sterile, is constant along the dottedlines (with the value indicated on the adjacent dashed lines).

pull σ8 down slightly to σ8 = 0.84 ± 0.01 and provides an ac-ceptable fit to the Planck data. For Planck TT+lowP+lensing,the difference in χ2 between the best fit base ΛCDM model andthe extension with ∆Neff = 0.39 is only ∆χ2

CMB ≈ 2. The higherspectral index with ∆Neff = 0.39 gives a decrease in large-scalepower, fitting the low ` < 30 Planck TT spectrum better by∆χ2 ≈ 1, but at the same time the high-` data prefer ∆Neff ≈ 0.Correlations with other cosmological parameters can be seenin Fig. 20. Clearly, a very effective way of testing these mod-els would be to obtain reliable, accurate, astrophysical measure-ments of H0 and σ8.

In summary, models with ∆Neff = 1 are disfavoured byPlanck combined with BAO data at about the 3σ level. Modelswith fractional changes of ∆Neff ≈ 0.39 are mildly disfavouredby Planck, but require higher H0 and σ8 compared to baseΛCDM.

6.4.3. Simultaneous constraints on Neff and neutrino mass

As discussed in the previous sections, neither a higher neu-trino mass nor additional radiation density alone can resolveall of the tensions between Planck and other astrophysical data.However, the presence of additional massive particles, such asmassive sterile neutrinos, could potentially improve the situa-tion by introducing enough freedom to allow higher values ofthe Hubble constant and lower values of σ8. As mentionedin Sect. 6.4.2, massive sterile neutrinos offer a possible solu-tion to reactor neutrino oscillation anomalies (Kopp et al. 2013;Giunti et al. 2013) and this has led to significant recent in-terest in this class of models (Hamann & Hasenkamp 2013;Wyman et al. 2014; Battye & Moss 2014; Leistedt et al. 2014;

Bergstrom et al. 2014; MacCrann et al. 2015). Alternatively, ac-tive neutrinos could have significant degenerate masses abovethe minimal baseline value together with additional masslessparticles contributing to Neff . Many more complicated scenarioscould also be envisaged.

In the case of massless radiation density, the cosmologicalpredictions are independent of the actual form of the distribu-tion function, since all particles travel at the speed of light.However, for massive particles the results are more model de-pendent. To formulate a well-defined model, we follow PCP13and consider the case of one massive sterile neutrino parameter-ized by meff

ν, sterile ≡ (94.1 Ων,sterileh2) eV, in addition to the twoapproximately massless and one massive neutrino of the base-line model. For thermally-distributed sterile neutrinos, meff

ν, sterileis related to the true mass via

meffν, sterile = (Ts/Tν)3mthermal

sterile = (∆Neff)3/4mthermalsterile , (63)

and for the cosmologically-equivalent Dodelson-Widrow (DW)case (Dodelson & Widrow 1994) the relation is given by

meffν, sterile = χs mDW

sterile , (64)

with ∆Neff = χs. We impose a prior on the physical thermalmass, mthermal

sterile < 10 eV, when generating parameter chains, toexclude regions of parameter space in which the particles areso massive that their effect on the CMB spectra is identical tothat of cold dark matter. Although we consider only the specificcase of one massive sterile neutrino with a thermal (or DW) dis-tribution, our constraints will be reasonably accurate for othermodels, for example eV-mass particles produced as non-thermaldecay products (Hasenkamp 2014).

Figure 32 shows that although Planck is perfectly consistentwith no massive sterile neutrinos, a significant region of param-eter space with fractional ∆Neff is allowed, where σ8 is lowerthan in the base ΛCDM model. This is also the case for masslesssterile neutrinos combined with massive active neutrinos. In thesingle massive sterile model, the combined constraints are

Neff < 3.7

meffν, sterile < 0.52 eV

95%, Planck TT+lowP+lensing+BAO.

(65)The upper tail of meff

ν, sterile is largely associated with high physicalmasses near to the prior cutoff; if instead we restrict to the regionwhere mthermal

sterile < 2 eV the constraint is

Neff < 3.7

meffν, sterile < 0.38 eV


(66)Massive sterile neutrinos with mixing angles large enough tohelp resolve the reactor anomalies would typically imply fullthermalization in the early Universe, and hence give ∆Neff = 1for each additional species. Such a high value of Neff , espe-cially combined with msterile ≈ 1 eV, as required by reactoranomaly solutions, were virtually ruled out by previous cos-mological data (Mirizzi et al. 2013; Archidiacono et al. 2013a;Gariazzo et al. 2013). This conclusion is strengthened by theanalysis presented here, since Neff = 4 is excluded at greaterthan 99 % confidence. Unfortunately, there does not appear to bea consistent resolution to the reactor anomalies, unless thermal-ization of the massive neutrinos can be suppressed, for example,by large lepton asymmetry, new interactions, or particle decay(see Gariazzo et al. 2014; Bergstrom et al. 2014, and referencestherein).

44


0.7

0.8

0.9

σ8

+Neff

0.24 0.32 0.40 0.48

Ωm

0.7

0.8

0.9

σ8

+Σmν

+Neff +Σmν

0.24 0.32 0.40 0.48

Ωm

+Neff +meffν, sterile

+Neff

56 64 72

H0

+Σmν

+Neff +Σmν

56 64 72

H0

+Neff +meffν, sterile

Planck TT+lowP +lensing +lensing+BAO ΛCDM

Fig. 33. 68 % and 95 % constraints from Planck TT+lowP (green), Planck TT+lowP+lensing (grey), and PlanckTT+lowP+lensing+BAO (red) on the late-Universe parameters H0, σ8, and Ωm in various neutrino extensions of the base ΛCDMmodel. The blue contours show the base ΛCDM constraints from Planck TT+lowP+lensing+BAO. The dashed cyan contoursshow joint constraints from the H13 CFHTLenS galaxy weak lensing likelihood (with angular cuts as in Fig. 18) at constant CMBacoustic scale θMC (fixed to the Planck TT+lowP ΛCDM best fit) combined with BAO and the Hubble constant measurement ofEq. (30). These additional constraints break large parameter degeneracies in the weak lensing likelihood that would otherwise ob-scure the comparison with the Planck contours. Here priors on other parameters applied to the CFHTLenS analysis are as describedin Sect. 5.5.2.

We have also considered the case of additional radiation anddegenerate massive active neutrinos, with the combined con-straint

Neff = 3.2 ± 0.5∑mν < 0.32 eV


(67)Again Planck shows no evidence for a deviation from the baseΛCDM model.

6.4.4. Neutrino models and tension with external data

The extended models discussed in this section allow Planck to beconsistent with a wider range of late-Universe parameters than inbase ΛCDM. Figure 33 summarizes the constraints on Ωm, σ8,and H0 for the various models that we have considered. The in-ferred Hubble parameter can increase or decrease, as required tomaintain the observed acoustic scale, depending on the relativecontribution of additional radiation (changing the sound hori-zon) and neutrino mass (changing mainly the angular diameterdistance). However, all of the models follow similar degeneracydirections in the Ωm–σ8 and H0–σ8 planes, so these models re-main predictive: large common areas of the parameter space areexcluded in all of these models. The two-parameter extensionsare required to fit substantially lower values of σ8 without alsodecreasing H0 below the values determined from direct measure-ments, but the scope for doing this is clearly limited.

External data sets need to be reanalysed consistently in ex-tended models, since the extensions change the growth of struc-ture, angular distances, and the matter-radiation equality scale.

For example, the dashed lines in Fig. 33 show how differentmodels affect the CFHTLenS galaxy weak lensing constraintsfrom Heymans et al. (2013) (see Sect. 5.5.2), when restrictedto the region of parameter space consistent with the Planckacoustic scale measurements and the local Hubble parameter.The filled green, grey, and red contours in Fig. 33 show theCMB constraints on these models for various data combina-tions. The tightest of these constraints comes from the PlanckTT+lowP+lensing+BAO combination. The blue contours showthe constraints in the base ΛCDM cosmology. The red contoursare broader than the blue contours and there is greater overlapwith the CFHTLenS contours, but this offers only a marginalimprovement compared to base ΛCDM (compare with Fig. 18;see also the discussions in Leistedt et al. 2014 and Battye et al.2015). For each of these models, the CFHTLenS results preferlower values of σ8. Allowing for a higher neutrino mass low-ers σ8 from Planck, but does not help alleviate the discrepancywith the CFHTLenS data, since the Planck data prefer a lowervalue of H0. A joint analysis of the CFHTLenS likelihood withPlanck TT+lowP shows a ∆χ2 < 1 preference for the extendedneutrino models compared to base ΛCDM, and the fits to PlanckTT+lowP are worse in all cases. In base ΛCDM the CFHTLenSdata prefer a region of parameter space ∆χ2 ≈ 4 away from thePlanck TT+lowP+CFHTLenS joint fit, indicative of the tensionbetween the data sets. This is only slightly relieved to ∆χ2 ≈ 3in the extended models.

In summary, modifications to the neutrino sector alone can-not easily explain the discrepancies between Planck and otherastrophysical data described in Sect. 5.5, including the inferenceof a low value of σ8 from rich cluster counts.

45


6.4.5. Testing perturbations in the neutrino background

As shown in the previous sections, the Planck data provide ev-idence for a cosmic neutrino background at a very high signifi-cance level. Neutrinos affect the CMB anisotropies at the back-ground level, by changing the expansion rate before recombina-tion and hence relevant quantities such as the sound horizon andthe damping scales. Neutrinos also affect the CMB anisotropiesvia their perturbations. Perturbations in the neutrino backgroundare coupled through gravity to the perturbations in the pho-ton background, and can be described (for massless neutrinos)by the following set of equations (Hu 1998; Hu et al. 1999;Trotta & Melchiorri 2005; Archidiacono et al. 2011):

δν =aa

(1 − 3c2

eff

) (δν + 3

aa

qνk

)− k

(qν +

23k

h)

; (68a)

qν = k c2eff

(δν + 3

aa

qνk

)−

aa

qν −23

kπν ; (68b)

πν = 3k c2vis

(25

qν +4

15k(h + 6η)

)−

35

kFν,3 ; (68c)

Fν,` =k

2` + 1(`Fν,`−1 − (` + 1) Fν,`+1

), (` ≥ 3) . (68d)

Here dots denote derivatives with respect to conformal time, δνis the neutrino density contrast, qν is the neutrino velocity pertur-bation, πν the anisotropic stress, Fν,` are higher-order momentsof the neutrino distribution function, and h and η are the scalarmetric perturbations in the synchronous gauge. In these equa-tions, c2

effis the neutrino sound speed in its own reference frame

and c2vis parameterizes the anisotropic stress. For standard non-

interacting massless neutrinos c2eff

= c2vis = 1/3. Any deviation

from the expected values could provide a hint of non-standardphysics in the neutrino sector.

A greater (lower) neutrino sound speed would increase (de-crease) the neutrino pressure, leading to a lower (higher) per-turbation amplitude. On the other hand, changing c2

vis alters theviscosity of the neutrino fluid. For c2

vis = 0, the neutrinos act asa perfect fluid, supporting undamped acoustic oscillations.

Several previous studies have used this approach toconstrain c2

effand c2

vis using cosmological data (see, e.g.,Trotta & Melchiorri 2005; Smith et al. 2012; Archidiacono et al.2013b; Gerbino et al. 2013; Audren et al. 2015), with the moti-vation that deviations from the expected values could be a hintof non-standard physics in the neutrino sector. Non-standard in-teractions could involve, for example, neutrino coupling withlight scalar particles (Hannestad 2005; Beacom et al. 2004; Bell2005; Sawyer 2006). If neutrinos are strongly coupled at recom-bination, this would result in a lower value for c2

vis than in thestandard model. Alternatively, the presence of early dark en-ergy that mimics a relativistic component at recombination couldpossibly lead to a value for c2

effthat differs from 1/3 (see, e.g.,

Calabrese et al. 2011).In this analysis, for simplicity, we assume Neff = 3.046 and

massless neutrinos. By using an equivalent parameterization formassive neutrinos (Audren et al. 2015) we have checked that as-suming one massive neutrino with Σmν ≈ 0.06 eV, as in the basemodel used throughout this paper, has no impact on the con-straints on c2

effand c2

vis reported in this section.29 We adopt a flatprior between zero and unity for both c2

vis and c2eff

.

29We also do not explore extended cosmologies in this section,since no significant degeneracies are expected between (

∑mν, Neff , w,

dns/d ln k) and (c2eff

, c2vis) (Audren et al. 2015).

0.28 0.30 0.32 0.34c2

eff

0.0

0.2

0.4

0.6

0.8

1.0

P/

Pm

ax

Planck TT+lowP

+BAO


+BAO

0.2 0.4 0.6 0.8 1.0c2

vis

0.0

0.2

0.4

0.6

0.8

1.0

P/

Pm

ax

Planck TT+lowP

+BAO


+BAO

Fig. 34. 1D posterior distributions for the neutrino perturbation param-eters c2

eff(top) and c2

vis (bottom). Dashed vertical lines indicate the con-ventional values c2

eff= c2

vis = 1/3.

The top and bottom panels of Fig. 34 show the pos-terior distributions of c2

effand c2

vis from Planck TT+lowP,Planck TT+lowP+BAO, Planck TT,TE,EE+lowP, and PlanckTT,TE,EE+lowP+BAO. The mean values and 68 % errors on c2

eff

and c2vis are

c2eff = 0.312 ± 0.011

c2vis = 0.47+0.26

−0.12

Planck TT+lowP,

(69a)c2

eff = 0.316 ± 0.010

c2vis = 0.44+0.15

−0.10

Planck TT+lowP+BAO,

(69b)c2

eff = 0.3240 ± 0.0060

c2vis = 0.327 ± 0.037

Planck TT,TE,EE+lowP,

(69c)c2

eff = 0.3242 ± 0.0059

c2vis = 0.331 ± 0.037

Planck TT,TE,EE+lowP+BAO.

(69d)

46


Constraints on these parameters are consistent with the con-ventional values c2

eff= c2

vis = 1/3. A vanishing value of c2vis,

which might imply a strong interaction between neutrinos andother species, is excluded at more than the 95 % level arisingfrom the Planck temperature data. This conclusion is greatlystrengthened (to about 9σ) when Planck polarization data areincluded. As discussed in Bashinsky & Seljak (2004), neutrinoanisotropic stresses introduce a phase shift in the CMB angularpower spectra, which is more visible in polarization than temper-ature because of the sharper acoustic peaks. This explains whywe see such a dramatic reduction in the error on c2

vis when in-cluding polarization data.

The precision of our results is consistent with the forecastsdiscussed in Smith et al. (2012), and we find strong evidence,purely from CMB observations, for neutrino anisotropies withthe standard values c2

vis = 1/3 and c2eff

= 1/3.

6.5. Primordial nucleosynthesis

6.5.1. Details of analysis approach

Standard big bang nucleosynthesis (BBN) predicts light elementabundances as a function of parameters relevant to the CMB,such as the baryon-to-photon density ratio ηb ≡ nb/nγ, the radi-ation density parameterized by Neff , and the chemical potentialof the electron neutrinos. In PCP13, we presented consistencychecks between the Planck 2013 results, light element abun-dance data, and standard BBN. The goal of Sect. 6.5.2 belowis to update these results and to provide improved tests of thestandard BBN model. In Sect. 6.5.3 we show how Planck datacan be used to constrain nuclear reaction rates, and in Sect. 6.5.4we will present the most stringent CMB bounds to date on theprimordial helium fraction.

For simplicity, our analysis assumes a negligible leptonicasymmetry in the electron neutrino sector. For a fixed photontemperature today (which we take to be T0 = 2.7255 K), ηb canbe related to ωb ≡ Ωbh2, up to a small (and negligible) uncer-tainty associated with the primordial helium fraction. StandardBBN then predicts the abundance of each light element as a func-tion of only two parameters, ωb and ∆Neff ≡ Neff − 3.046, witha theoretical error coming mainly from uncertainties in the neu-tron lifetime and a few nuclear reaction rates.

We will confine our discussion to BBN predictions for theprimordial abundances30 of 4He and deuterium, expressed, re-spectively as YBBN

P = 4nHe/nb and yDP = 105nD/nH. We willnot discuss other light elements, such as tritium and lithium, be-cause the observed abundance measurements and their interpre-tation is more controversial (see Fields et al. 2014, for a recentreview). As in PCP13, the BBN predictions for YBBN

P (ωb,∆Neff)and yDP(ωb,∆Neff) are given by Taylor expansions obtained withthe PArthENoPE code (Pisanti et al. 2008), similar to the onespresented in Iocco et al. (2009), but updated by the PArthENoPEteam with the latest observational data on nuclear rates and on

30BBN calculations usually refer to nucleon number density frac-tions rather than mass fractions. To avoid any ambiguity with the heliummass fraction YP, normally used in CMB physics, we use superscriptsto distinguish between the two definitions YCMB

P and YBBNP . Typically,

YBBNP is about 0.5 % higher than YCMB

P .

0.24

0.25

0.26

YB

BN

P

Aver et al. (2013)

Standard BBN

0.018 0.020 0.022 0.024 0.026ωb

2.2

2.6

3.0

3.4

y DP

Iocco et al. (2008)

Cooke et al. (2014)

PlanckTT+lowP+BAO

Fig. 35. Predictions of standard BBN for the primordial abun-dance of 4He (top) and deuterium (bottom), as a function of thebaryon density ωb. The width of the green stripes correspondsto 68 % uncertainties on nuclear reaction rates and on the neu-tron lifetime. The horizontal bands show observational boundson primordial element abundances compiled by various authors,and the red vertical band shows the Planck TT+lowP+BAObounds on ωb (all with 68 % errors). The BBN predictions andCMB results shown here assume Neff = 3.046 and no significantlepton asymmetry.

the neutron life-time:

YBBNP = 0.2311 + 0.9502ωb − 11.27ω2

b

+ ∆Neff

(0.01356 + 0.008581ωb − 0.1810ω2

b

)+ ∆N2

eff

(−0.0009795 − 0.001370ωb + 0.01746ω2

b

);

(70)

yDP = 18.754 − 1534.4ωb + 48656ω2b − 552670ω3

b

+ ∆Neff

(2.4914 − 208.11ωb + 6760.9ω2

b − 78007ω3b

)+ ∆N2

eff

(0.012907 − 1.3653ωb + 37.388ω2

b − 267.78ω3b

).

(71)

By averaging over several measurements, the Particle DataGroup 2014 (Olive et al. 2014) estimates the neutron life-timeto be τn = (880.3 ± 1.1) s at 68 % CL.31 The expansions inEqs. (70) and (71) are based on this central value, and we as-sume that Eq. (70) predicts the correct helium fraction up to astandard error σ(YBBN

P ) = 0.0003, obtained by propagating theerror on τn.

The uncertainty on the deuterium fraction is dominated bythat on the rate of the reaction d(p, γ)3He. For that rate, inPCP13 we relied on the result of Serpico et al. (2004), obtainedby fitting several experiments. The expansions of Eqs. (70)and (71) now adopt the latest experimental determination byAdelberger et al. (2011) and use the best-fit expression in their

31However, the most recent individual measurement by Yue et al.(2013) gives τn = [887.8 ± 1.2 (stat.) ± 1.9 (syst.)] s, which is dis-crepant at 3.3σ with the previous average (including only statisticalerrors). Hence one should bear in mind that systematic effects could beunderestimated in the Particle Data Group result. Adopting the centralvalue of Yue et al. (2013) would shift our results by a small amount (bya factor of 1.0062 for YP and 1.0036 for yDP).

47


equation (29). We also rely on the uncertainty quoted inAdelberger et al. (2011) and propagate it to the deuterium frac-tion. This gives a standard error σ(yDP) = 0.06, which is moreconservative than the error adopted in PCP13.

6.5.2. Primordial abundances from Planck data andstandard BBN

We first investigate the consistency of standard BBN and theCMB by fixing the radiation density to its standard value, i.e.,Neff = 3.046, based on the assumption of standard neutrinodecoupling and no extra light relics. We can then use Planckdata to measure ωb, assuming base ΛCDM, and test for consis-tency with experimental abundance measurements. The 95 % CLbounds obtained for the base ΛCDM model for various datacombinations are

ωb =

0.02222+0.00045

−0.00043 Planck TT+lowP,

0.02226+0.00040−0.00039 Planck TT+lowP+BAO,

0.02225+0.00032−0.00030 Planck TT,TE,EE+lowP,

0.02229+0.00029−0.00027 Planck TT,TE,EE+lowP+BAO,

(72)corresponding to a predicted primordial 4He number densityfraction (95 % CL) of

YBBNP =

0.24665+(0.00020) 0.00063

−(0.00019) 0.00063 Planck TT+lowP,

0.24667+(0.00018) 0.00063−(0.00018) 0.00063 Planck TT+lowP+BAO,

0.24667+(0.00014) 0.00062−(0.00014) 0.00062 Planck TT,TE,EE+lowP,

0.24668+(0.00013) 0.00061−(0.00013) 0.00061 Planck TT,TE,EE+lowP+BAO,

(73)and deuterium fraction (95 % CL)

yDP =

2.620+(0.083) 0.15

−(0.085) 0.15 Planck TT+lowP,

2.612+(0.075) 0.14−(0.074) 0.14 Planck TT+lowP+BAO,

2.614+(0.057) 0.13−(0.060) 0.13 Planck TT,TE,EE+lowP,

2.606+(0.051) 0.13−(0.054) 0.13 Planck TT,TE,EE+lowP+BAO.

(74)The first set of error bars (in parentheses) in Eqs. (73) and (74)reflect only the uncertainty on ωb. The second set includes thetheoretical uncertainty on the BBN predictions, added in quadra-ture to the errors from ωb. The total errors in the predicted he-lium abundances are dominated by the BBN uncertainty, as inPCP13. For deuterium, the Planck 2015 results improve the de-termination of ωb to the point where the theoretical errors arecomparable or larger than the errors from the CMB. In otherwords, for base ΛCDM the predicted abundances cannot be im-proved substantially by further measurements of the CMB. Thisalso means that Planck results can, in principle, be used to in-vestigate nuclear reaction rates that dominate the theoretical un-certainty (see Sect. 6.5.3).

The results of Eqs. (73) and (74) are well within theranges indicated by the latest measurements of primordial abun-dances, as illustrated in Fig. 35. The helium data compilationof Aver et al. (2013) gives YBBN

P = 0.2465 ± 0.0097 (68 % CL),and the Planck prediction is near the middle of this range.32 As

32A substantial part of this error comes from the regression to zerometallicity. The mean of the 17 measurements analysed by Aver et al.(2013) is 〈YBBN

P 〉 = 0.2535 ± 0.0036, i.e., about 1.7σ higher than thePlanck predictions of Eq. (73).

0.018 0.020 0.022 0.024 0.026ωb

01

23

45

67

8

Neff

Aver et al. (2013)

Cooke et al. (2014)

Planck TT+lowP

Planck TT+lowP+BAO


Fig. 36. Constraints in the ωb–Neff plane from Planck andPlanck+BAO data (68 % and 95 % contours) compared to thepredictions of BBN, given primordial element abundance mea-surements. We show the 68 % and 95 % confidence regions de-rived from 4He bounds compiled by Aver et al. (2013) and fromdeuterium bounds compiled by Cooke et al. (2014). In the CMBanalysis, Neff is allowed to vary as an additional parameter tobase ΛCDM, with YP fixed as a function of ωb and Neff , accord-ing to BBN predictions. These constraints assume no significantlepton asymmetry.

summarized by Aver et al. (2013) and Peimbert (2008), heliumabundance measurements derived from emission lines in low-metallicity H ii regions are notoriously difficult and prone to sys-tematic errors. As a result, many discrepant helium abundancemeasurements can be found in the literature. Izotov et al. (2014)have reported YBBN

P = 0.2551± 0.0022, which is discrepant withthe base ΛCDM predictions by 3.4σ. Such a high helium frac-tion could be accommodated by increasing Neff (see Fig. 36 andSect. 6.5.4); however, at present it is not clear whether the er-ror quoted by Izotov et al. (2014) accurately reflects systematicuncertainties, including in particular the error in extrapolating tozero metallicity.

Historically, deuterium abundance measurements haveshown excess scatter over that expected from statistical errors,indicating the presence of systematic uncertainties in the ob-servations. Figure 35 shows the data compilation of Iocco et al.(2009), yDP = 2.87 ± 0.22 (68 % CL), which includes mea-surements based on damped Lyα and Lyman limit systems.We also show the more recent results by Cooke et al. (2014)(see also Pettini & Cooke 2012) based on their observations oflow-metallicity damped Lyα absorption systems in two quasars(SDSS J1358+6522, zabs = 3.06726 and SDSS J1419+0829,zabs = 3.04973) and a reanalysis of archival spectra of dampedLyα systems in three further quasars that satisfy strict selectioncriteria. The Cooke et al. (2014) analysis gives yDP = 2.53±0.04(68 % CL), somewhat lower than the central Iocco et al. (2009)value, and with a much smaller error. The Cooke et al. (2014)value is almost certainly the more reliable measurement, as ev-idenced by the consistency of the deuterium abundances of thefive systems in their analysis. The Planck base ΛCDM predic-tions of Eq. (74) lie within 1σ of the Cooke et al. (2014) result.This is a remarkable success for the standard theory of BBN.

48


It is worth noting that the Planck data are so accurate that ωbis insensitive to the underlying cosmological model. In our gridof extensions to base ΛCDM the largest degradation of the errorin ωb is in models that allow Neff to vary. In these models, themean value of ωb is almost identical to that for base ΛCDM, butthe error on ωb increases by about 30 %. The value of ωb is sta-ble to even more radical changes to the cosmology, for example,adding general isocurvature modes (Planck Collaboration XX2016).

If we relax the assumption that Neff = 3.046 (but adhereto the hypothesis that electron neutrinos have a standard distri-bution, with a negligible chemical potential), BBN predictionsdepend on both parameters (ωb and Neff . Following the samemethodology as in Sect. 6.4.4 of PCP13, we can identify the re-gion of the ωb–Neff parameter space that is compatible with di-rect measurements of the primordial helium and deuterium abun-dances, including the BBN theoretical errors. This is illustratedin Fig. 36 for the Neff extension to base ΛCDM. The region pre-ferred by CMB observations lies at the intersection between thehelium and deuterium abundance 68 % CL preferred regions andis compatible with the standard value of Neff = 3.046. This con-firms the beautiful agreement between CMB and BBN physics.Figure 36 also shows that the Planck polarization data help inreducing the degeneracy between ωb and Neff .

We can actually make a more precise statement by combin-ing the posterior distribution on ωb and Neff) obtained for Planckwith that inferred from helium and deuterium abundance, in-cluding observational and theoretical errors. This provides jointCMB+BBN predictions on these parameters. After marginaliz-ing over ωb, the 95 % CL preferred ranges for Neff are

Neff =

3.11+0.59

−0.57 He+Planck TT+lowP,

3.14+0.44−0.43 He+Planck TT+lowP+BAO,

2.99+0.39−0.39 He+Planck TT,TE,EE+lowP,

(75)

when combining Planck with the helium abundance estimatedby Aver et al. (2013), or

Neff =

2.95+0.52

−0.52 D+Planck TT+lowP,

3.01+0.38−0.37 D+Planck TT+lowP+BAO,

2.91+0.37−0.37 D+Planck TT,TE,EE+lowP,

(76)

when combining with the deuterium abundance measuredby Cooke et al. (2014). These bounds represent the bestcurrently-available estimates of Neff and are remarkably consis-tent with the standard model prediction.

The allowed region in ωb–Neff space does not increase sig-nificantly when other parameters are allowed to vary at the sametime. From our grid of extended models, we have checked thatthis conclusion holds in models with neutrino masses, tensorfluctuations, or running of the scalar spectral index, for exam-ple.

6.5.3. Constraints from Planck and deuterium observationson nuclear reaction rates

We have seen that primordial element abundances estimatedfrom direct observations are consistent with those inferred fromPlanck data under the assumption of standard BBN. However,the Planck determination of ωb is so precise that the theoreti-cal errors in the BBN predictions are now a dominant sourceof uncertainty. As noted by Cooke et al. (2014), one can beginto think about using CMB measurements together with accurate

deuterium abundance measurements to learn about the underly-ing BBN physics.

While for helium the theoretical error comes mainly fromthe uncertainties in the neutron lifetime, for deuterium it isdominated by uncertainties in the radiative capture processd(p, γ)3He, converting deuterium into helium. The present ex-perimental uncertainty for the S -factor at low energy (relevantfor BBN), is in the range 6–10 % (Ma et al. 1997). However,as noted by several authors (see, e.g., Nollett & Holder 2011;Di Valentino et al. 2014) the best-fit value of S (E) inferred fromexperimental data in the range 30 keV≤ E ≤ 300 keV is about5–10 % lower than theoretical expectations (Viviani et al. 2000;Marcucci et al. 2005). The PArthENoPE BBN code assumes thelower experimental value for d(p, γ)3He, and this might explainwhy the deuterium abundance determined by Cooke et al. (2014)is slightly lower than the value inferred by Planck.

To investigate this further, following the methodology ofDi Valentino et al. (2014), we perform a combined analysis ofPlanck and deuterium observations, to constrain the value of thed(p, γ)3He reaction rate. As in Di Valentino et al. (2014), we pa-rameterize the thermal rate R2(T ) of the d(p, γ)3He process inthe PArthENoPE code by introducing a rescaling factor A2 ofthe experimental rate R ex

2 (T ), i.e., R2(T ) = A2 Rex2 (T ), and solve

for A2 using various Planck+BAO data combinations, given theCooke et al. (2014) deuterium abundance measurements.

Assuming the base ΛCDM model we find (68 % CL)

A2 = 1.106 ± 0.071 Planck TT+lowP , (77a)A2 = 1.098 ± 0.067 Planck TT+lowP+BAO , (77b)A2 = 1.110 ± 0.062 Planck TT,TE,EE+lowP , (77c)A2 = 1.109 ± 0.058 Planck TT,TE,EE+lowP+BAO . (77d)

The posteriors for A2 are shown in Fig. 37. These results sug-gest that the d(p, γ)3He reaction rate may be have been under-estimated by about 10 %. Evidently, tests of the standard BBNpicture appear to have reached the point where they are limitedby uncertainties in nuclear reaction rates. There is therefore astrong case to improve the precision of experimental measure-ments (e.g., Anders et al. 2014) and theoretical computations ofkey nuclear reaction rates relevant for BBN.

6.5.4. Model-independent bounds on the helium fractionfrom Planck

Instead of inferring the primordial helium abundance from BBNcodes using (ωb,Neff) constraints from Planck, we can measure itdirectly, since variations in YBBN

P modify the density of free elec-trons between helium and hydrogen recombination and thereforeaffect the damping tail of the CMB anisotropies.

If we allow YCMBP to vary as an additional parameter to base

ΛCDM, we find the following constraints (at 95 % CL):

YBBNP =

0.253+0.041

−0.042 Planck TT+lowP ;

0.255+0.036−0.038 Planck TT+lowP+BAO ;

0.251+0.026−0.027 Planck TT,TE,EE+lowP ;

0.253+0.025−0.026 Planck TT,TE,EE+lowP+BAO .

(78)Joint constraints on YBBN

P and ωb are shown in Fig. 38. Theaddition of Planck polarization measurements results in a sub-stantial reduction in the uncertainty on the helium fraction.In fact, the standard deviation on YBBN

P in the case of PlanckTT,TE,EE+lowP is only 30 % larger than the observational er-ror quoted by Aver et al. (2013). As emphasized throughout this

49


0.90 1.05 1.20 1.35 1.50

A2

0.0

0.2

0.4

0.6

0.8

1.0

P/

Pm

ax

Planck TT+lowP+BBN

+BAO

Planck TT,TE,EE+lowP+BBN

+BAO

Fig. 37. Posteriors for the A2 reaction rate parameter for vari-ous data combinations. The vertical dashed line shows the valueA2 = 1 that corresponds to the current experimental estimate ofthe d(p, γ)3He rate used in the PArthENoPE BBN code.

0.020 0.022 0.024 0.026ωb

0.15

0.20

0.25

0.30

0.35

YB

BN

P

Aver et al. (2013)

Excluded by

Serenelli & Basu (2010)

Standard BBN

Planck TT+lowP

Planck TT+lowP+BAO


Fig. 38. Constraints in the ωb–YBBNP plane from Planck and

Planck+BAO, compared to helium abundance measurements.Here 68 % and 95 % contours are plotted for the CMB(+BAO)data combinations when YBBN

P is allowed to vary as an additionalparameter to base ΛCDM. The horizontal band shows observa-tional bounds on 4He compiled by Aver et al. (2013) with 68 %and 95 % errors, while the dashed line at the top of the figuredelineates the conservative 95 % upper bound inferred from theSolar helium abundance by Serenelli & Basu (2010). The greenstripe shows the predictions of standard BBN for the primordialabundance of 4He as a function of the baryon density. Both BBNpredictions and CMB results assume Neff = 3.046 and no signif-icant lepton asymmetry.

paper, the systematic effects in the Planck polarization spectra,although at low levels, have not been accurately characterizedat this time. Readers should therefore treat the polarization con-straints with some caution. Nevertheless, as shown in Fig. 38,all three data combinations agree well with the observed helium

0 1 2 3 4 5Neff

0.15

0.20

0.25

0.30

0.35

YB

BN

P Aver et al. (2013)

Excluded by

Serenelli & Basu (2010)Standard BBN

Planck TT+lowP

Planck TT+lowP+BAO


Fig. 39. As in Fig. 38, but now allowing YBBNP and Neff to vary as

parameter extensions to base ΛCDM.

abundance measurements and with the predictions of standardBBN.

There is a well-known parameter degeneracy between YPand the radiation density (see the discussion in PCP13). Heliumabundance predictions from the CMB are therefore particularlysensitive to the addition of the parameter Neff to base ΛCDM.Allowing both YBBN

P and Neff to vary we find the following con-straints (at 95 % CL):

YBBNP =

0.252+0.058

−0.065 Planck TT+lowP ;

0.251+0.058−0.064 Planck TT+lowP+BAO ;

0.263+0.034−0.037 Planck TT,TE,EE+lowP ;

0.262+0.035−0.037 Planck TT,TE,EE+lowP+BAO .

(79)Contours in the YBBN

P –Neff plane are shown in Fig. 39. Hereagain, the impact of Planck polarization data is important, andhelps to substantially reduce the degeneracy between these twoparameters. The Planck TT,TE,EE+lowP contours are in verygood agreement with standard BBN and Neff = 3.046. However,even if we relax the assumption of standard BBN, the CMB doesnot allow high values of Neff . It is therefore difficult to accommo-date an extra thermalized relativistic species, even if the standardBBN prior on the helium fraction is relaxed.

6.6. Dark matter annihilation

Energy injection from dark matter (DM) annihilation canalter the recombination history, leading to changes in thetemperature and polarization power spectra of the CMB(e.g., Chen & Kamionkowski 2004; Padmanabhan & Finkbeiner2005). As demonstrated in several papers (e.g., Galli et al. 2009;Slatyer et al. 2009; Finkbeiner et al. 2012), CMB anisotropiesoffer an opportunity to constrain the nature of DM. Furthermore,CMB experiments such as Planck can achieve limits on the an-nihilation cross-section that are relevant for the interpretation ofthe rise in the cosmic-ray positron fraction at energies >∼ 10 GeVobserved by PAMELA, Fermi, and AMS (Adriani et al. 2009;Ackermann et al. 2012; Aguilar et al. 2014). The CMB con-straints are complementary to those determined from other astro-

50


physical probes, such as the γ-ray observations of dwarf galaxiesby the Fermi satellite (Ackermann et al. 2014).

The way in which DM annihilations heat and ionize thegaseous background depends on the nature of the cascade of par-ticles produced following annihilation and, in particular, on theproduction of e± pairs and photons that couple to the gas. Thefraction of the rest mass energy that is injected into the gas can bemodelled by an “efficiency factor,” f (z), which is typically in therange 0.01–1 and depends on redshift.33 Computations of f (z)for various annihilation channels can be found in Slatyer et al.(2009), Hutsi et al. (2009), and Evoli et al. (2013). The rate ofenergy release per unit volume by annihilating DM can there-fore be written as

dEdtdV

(z) = 2 g ρ2critc

2Ω2c(1 + z)6 pann(z), (80)

where pann is defined as

pann(z) ≡ f (z)〈σ3〉

mχ. (81)

Here ρcrit the critical density of the Universe today, mχ is themass of the DM particle, and 〈σ3〉 is the thermally-averagedannihilation cross-section times the velocity (explicitly the so-called Møller velocity); we will refer to this quantity loosely asthe “cross-section” hereafter. In Eq. (80), g is a degeneracy fac-tor that is equal to 1/2 for Majorana particles and 1/4 for Diracparticles. In this paper, the constraints will refer to Majoranaparticles. Note that to produce the observed dark matter densityfrom thermal DM relics requires an annihilation cross-section of〈σ3〉 ≈ 3 × 10−26 cm3 s−1 (assuming s-wave annihilation) at thetime of freeze-out (see, e.g., the review by Profumo 2013).

Both the amplitude and redshift dependence of the effi-ciency factor f (z) depend on the details of the annihilation pro-cess (e.g., Slatyer et al. 2009). The functional shape of f (z)can be taken into account using generalized parameterizationsor principal components (Finkbeiner et al. 2012; Hutsi et al.2011), similar to the analysis of the recombination history pre-sented in Sect. 6.7.4. However, as shown in Galli et al. (2011),Giesen et al. (2012), and Finkbeiner et al. (2012), to a first ap-proximation the redshift dependence of f (z) can be ignored,since current CMB data (including Planck) are sensitive to en-ergy injection over a relatively narrow range of redshift, typi-cally z ≈ 1000–600. The effects of DM annihilation can there-fore be reasonably well parameterized by a single constant pa-rameter, pann (with f (z) set to a constant feff), which encodesthe dependence on the properties of the DM particles. In thefollowing, we calculate constraints on the pann parameter, as-suming that it is constant, and then project these constraints onto a particular dark matter model assuming feff ≡ f (z = 600),since the effect of dark matter annihilation peaks at z ≈ 600 (seeFinkbeiner et al. 2012). The f (z) functions used here are thosecalculated in Slatyer et al. (2009), with the updates described inGalli et al. (2013) and Madhavacheril et al. (2014). Finally, weestimate the fractions of injected energy that affect the gaseousbackground, from heating, ionizations, or Lyα excitations, us-ing the updated calculations described in Galli et al. (2013) andValdes et al. (2010), following Shull & van Steenberg (1985).

33To maintain consistency with other papers on dark matter annihi-lation, we retain the notation f (z) for the efficiency factor in this sec-tion; it should not be confused with the growth rate factor introduced inEq. (32).

0 2 4 6 8

pann [10−27cm3 s−1 GeV−1]

0.950

0.975

1.000

1.025

ns


Planck TE+lowP

Planck EE+lowP

Planck TT+lowP

WMAP9

Fig. 40. 2D marginal distributions in the pann–ns plane for PlanckTT+lowP (red), Planck EE+lowP (yellow), Planck TE+lowP(green), and Planck TT,TE,EE+lowP (blue) data combinations.We also show the constraints obtained using WMAP9 data (lightblue).

We compute the theoretical angular power spectrum in thepresence of DM annihilations by modifying the recfast routine(Seager et al. 1999) in the camb code as in Galli et al. (2011).34

We then add pann as an additional parameter to those of the baseΛCDM cosmology. Table 6 shows the constraints for variousdata combinations.

Table 6. Constraints on pann in units of cm3 s−1 GeV−1.

Data combinations pann (95 % upper limits)

TT+lowP . . . . . . . . . . . . . . . . . < 5.7 × 10−27

EE+lowP . . . . . . . . . . . . . . . . . < 1.4 × 10−27

TE+lowP . . . . . . . . . . . . . . . . . < 5.9 × 10−28

TT+lowP+lensing . . . . . . . . . . . < 4.4 × 10−27

TT,TE,EE+lowP . . . . . . . . . . . . < 4.1 × 10−28

TT,TE,EE+lowP+lensing . . . . . . < 3.4 × 10−28

TT,TE,EE+lowP+ext . . . . . . . . . < 3.5 × 10−28

The constraints on pann from the Planck TT+lowP spec-tra are about 3 times weaker than the 95 % limit of pann <2.1 × 10−27 cm3 s−1 GeV−1 derived from WMAP9, which in-cludes WMAP polarization data at low multipoles. On the otherhand, the Planck T E or EE spectra improve the constraints onpann by about an order of magnitude compared to those fromPlanck TT alone. This is because the main effect of dark matterannihilation is to increase the width of last scattering, leadingto a suppression of the amplitude of the peaks, both in tem-perature and polarization. As a result, the effects of DM an-nihilation on the power spectra at high multipole are degen-erate with other parameters of base ΛCDM, such as ns andAs (Chen & Kamionkowski 2004; Padmanabhan & Finkbeiner2005). At large angular scales (` . 200), however, dark matterannihilation can produce an enhancement in polarization, caused

34We checked that we obtain similar results using either the HyReccode (Ali-Haimoud & Hirata 2011), as detailed in Giesen et al. (2012),or CosmoRec (Chluba & Thomas 2011), instead of recfast.

51


1 10 100 1000 10000

mχ[GeV]

10−27

10−26

10−25

10−24

10−23

f eff〈σv〉[

cm3

s−1]

Thermal relic

Planck TT,TE,EE+lowPWMAP9CVLPossible interpretations for:AMS-02/Fermi/PamelaFermi GC

Fig. 41. Constraints on the self-annihilation cross-section at re-combination, 〈σ3〉z∗ , times the efficiency parameter, feff (Eq. 81).The blue area shows the parameter space excluded by the PlanckTT,TE,EE+lowP data at 95 % CL. The yellow line indicates theconstraint using WMAP9 data. The dashed green line delineatesthe region ultimately accessible to a cosmic-variance-limited ex-periment with angular resolution comparable to that of Planck.The horizontal red band includes the values of the thermal-relic cross-section multiplied by the appropriate feff for differ-ent DM annihilation channels. The dark grey circles show thebest-fit DM models for the PAMELA/AMS-02/Fermi cosmic-ray excesses, as calculated in Cholis & Hooper (2013) (captionof their figure 6). The light grey stars show the best-fit DM mod-els for the Fermi Galactic centre γ-ray excess, as calculated byCalore et al. (2015) (their tables I, II, and III), with the lightgrey area indicating the astrophysical uncertainties on the best-fit cross-sections.

by the increased ionization fraction in the freeze-out tail follow-ing recombination. As a result, large-angle polarization infor-mation is crucial for breaking the degeneracies between param-eters, as illustrated in Fig. 40. The strongest constraints on panntherefore come from the full Planck temperature and polariza-tion likelihood and there is little improvement if other astrophys-ical data, or Planck lensing, are added.35

We verified the robustness of the Planck TT,TE,EE+lowPconstraint by also allowing other parameter extensions of baseΛCDM (Neff , dns/d ln k, or YP) to vary together with pann.We found that the constraint is weakened by up to 20 %.Furthermore, we have verified that we obtain consistent resultswhen relaxing the priors on the amplitudes of the Galactic dusttemplates or if we use the CamSpec likelihood instead of thebaseline Plik likelihood.

Figure 41 shows the constraints from WMAP9, PlanckTT,TE,EE+lowP, and a forecast for a cosmic-variance-limitedexperiment with similar angular resolution to Planck.36 The hor-izontal red band includes the values of the thermal-relic cross-section multiplied by the appropriate feff for different DM anni-hilation channels. For example, the upper red line corresponds to

35It is interesting to note that the constraint derived from PlanckTT,TE,EE+lowP is consistent with the forecast given in Galli et al.(2009), pann < 3 × 10−28 cm3 s−1 GeV−1.

36We assumed here that the cosmic-variance-limited experimentwould measure the angular power spectra up to a maximum multipoleof `max = 2500, observing a sky fraction fsky = 0.65.

feff = 0.67, which is appropriate for a DM particle of mass mχ =10 GeV annihilating into e+e−, while the lower red line corre-sponds to feff = 0.13, for a DM particle annihilating into 2π+π−

through an intermediate mediator (see, e.g., Arkani-Hamed et al.2009). The Planck data exclude at 95 % confidence level a ther-mal relic cross-section for DM particles of mass mχ <∼ 44 Gevannihilating into e+e− ( feff ≈ 0.6), mχ <∼ 16 GeV annihilatinginto µ+µ− or bb ( feff ≈ 0.2), and mχ <∼ 11 GeV annihilating intoτ+τ− ( feff ≈ 0.15).

The dark grey shaded area in Fig. 41 shows the approx-imate allowed region of parameter space, as calculated byCholis & Hooper (2013) on the assumption that the PAMELA,AMS, and Fermi cosmic-ray excesses are caused by DM annihi-lation; the dark grey dots indicate the best-fit dark matter modelsdescribed in that paper (for a recent discussion on best-fittingmodels, see also Boudaud et al. 2015). The favoured value ofthe cross-section is about two orders of magnitude higher thanthe thermal relic cross-section (≈ 3×10−26 cm3 s−1). Attempts toreconcile such a high cross-section with the relic abundance ofDM include a Sommerfeld enhanced cross-section (that may sat-urate at 〈σ3〉 ≈ 10−24 cm3 s−1) or non-thermal production of DM(see, e.g., the discussion by Madhavacheril et al. 2014). Both ofthese possibilities are strongly disfavoured by the Planck data.We cannot, however, exclude more exotic possibilities, such asDM annihilation through a p-wave channel with a cross-sectionthat scales as 32 (Diamanti et al. 2014). Since the relative veloc-ity of DM particles at recombination is many orders of magni-tude smaller than in the Galactic halo, such a model cannot beconstrained using CMB data.

Observations from the Fermi Large Area Telescopeof extended γ-ray emission towards the centre of theMilky Way, peaking at energies of around 1–3 GeV, havebeen interpreted as evidence for annihilating DM (e.g.,Goodenough & Hooper 2009; Gordon & Macıas 2013;Daylan et al. 2016; Abazajian et al. 2014; Lacroix et al. 2014).The light grey stars in Fig. 41 show specific models of DMannihilation designed to fit the Fermi γ-ray excess (Calore et al.2015), while the light grey box shows the uncertainties ofthe best-fit cross-sections due to imprecise knowledge of theGalactic DM halo profile. Although the interpretation of theFermi excess remains controversial (because of uncertaintiesin the astrophysical backgrounds), DM annihilation remains apossible explanation. The best-fit models of Calore et al. (2015)are consistent with the Planck constraints on DM annihilation.

6.7. Testing recombination physics with Planck

The cosmological recombination process determines how CMBphotons decoupled from baryons around redshift z ≈ 103,when the Universe was about 400 000 years old. The impor-tance of this transition on the CMB anisotropies has long beenrecognized (Sunyaev & Zeldovich 1970; Peebles & Yu 1970).The most advanced computations of the ionization history(e.g., Ali-Haımoud & Hirata 2010; Chluba & Thomas 2011;Ali-Haimoud & Hirata 2011; Chluba et al. 2012) account formany subtle atomic physics and radiative transfer effects thatwere not included in the earliest calculations (Zeldovich et al.1968; Peebles 1968).

With precision data from Planck, we are sensitive to sub-percent variations of the free electron fraction around last-scattering (e.g., Hu et al. 1995; Seager et al. 2000; Seljak et al.2003). Quantifying the impact of uncertainties in the ionizationhistory around the maximum of the Thomson visibility function

52


on predictions of the CMB power spectra is thus crucial for thescientific interpretation of data from Planck. In particular, fortests of models of inflation and extensions to ΛCDM, the inter-pretation of the CMB data can be significantly compromised byinaccuracies in the recombination calculation (e.g., Wong et al.2008; Rubino-Martın et al. 2010; Shaw & Chluba 2011). Thisproblem can be approached in two ways, either by using mod-ified recombination models with a specific physical process (orparameter) in mind, or in a semi-blind, model-independent way.Both approaches provide useful insights in assessing the robust-ness of the results from Planck.

Model-dependent limits on varying fundamental constants(Kaplinghat et al. 1999; Scoccola et al. 2009; Galli et al. 2009),annihilating or decaying particles (e.g., Chen & Kamionkowski2004; Padmanabhan & Finkbeiner 2005; Zhang et al. 2006, andSect. 6.6), or more general sources of extra ionization and ex-citation photons (Peebles et al. 2000; Doroshkevich et al. 2003;Galli et al. 2008), have been discussed extensively in the litera-ture.

As already discussed in PCP13, the choice for Planck hasbeen to use the rapid calculations of the recfast code, mod-ified using corrections calculated with the more precise codes.To start this sub-section we quantify the effect on the analy-sis of Planck data of the remaining uncertainties in the stan-dard recombination history obtained with different recombina-tion codes (Sect. 6.7.1). We also derive CMB anisotropy-basedmeasurements of the hydrogen 2s–1s two-photon decay rate,A2s→1s (Sect. 6.7.2), and the average CMB temperature, T0 de-rived at the last-scattering epoch (Sect. 6.7.3). These two param-eters strongly affect the recombination history but are usuallykept fixed when fitting models to CMB data (as in the analysesdescribed in previous sections). Section 6.7.4 describes model-independent constraints on perturbed recombination scenarios.A discussion of these cases provides both a test of the consis-tency of the CMB data with the standard recombination scenarioand also a demonstration of the impressive sensitivity of Planckto small variations in the ionization history at z ≈ 1100.

6.7.1. Comparison of different recombination codes

Even for pre-Planck data, it was realized that the early recombi-nation calculations of Zeldovich et al. (1968) and Peebles (1968)had to be improved. This led to the development of the widely-used and computationally quick recfast code (Seager et al.1999, 2000). However, for Planck, the recombination model ofrecfast in its original form is not accurate enough. Percent-level corrections, due to detailed radiative transfer and atomicphysics effects have to be taken into account. Ignoring these ef-fects can bias the inferred cosmological parameters, some by asmuch as a few standard deviations.

The recombination problem was solved as a com-mon effort of several groups (Dubrovich & Grachev2005; Kholupenko et al. 2007; Chluba & Sunyaev 2006b;Rubino-Martın et al. 2006; Karshenboim & Ivanov 2008;Wong & Scott 2007; Switzer & Hirata 2008; Grin & Hirata2010; Ali-Haımoud & Hirata 2010). This work was undertaken,to a large extent, in preparation for the precision data fromPlanck. Both CosmoRec (Chluba & Thomas 2011) and HyRec(Ali-Haimoud & Hirata 2011) allow fast and precise computa-tions of the ionization history, explicitly capturing the physicsof the recombination problem. For the standard cosmology,the ionization histories obtained from these two codes intheir default settings agree to within 0.05 % for hydrogenrecombination (600 <∼ z <∼ 1600) and 0.35 % during helium

recombination37 (1600 <∼ z <∼ 3000). The effect of these smalldifferences on the CMB power spectra is <∼ 0.1 % at ` < 4000and so has a negligible impact on the interpretation of precisionCMB data; for the standard six parameters of base ΛCDM, wefind that the largest effect is a bias in ln(1010As) at the level of0.04σ ≈ 0.0012 for Planck TT,TE,EE+lowP+BAO.

For Planck analyses, the recombination model of recfast isused by default. In recfast, the precise dynamics of recombi-nation is not modelled physically, but approximated with fitting-functions calibrated against the full recombination calculationsassuming a reference cosmology (Seager et al. 1999, 2000;Wong et al. 2008). At the level of precision required for Planck,the recfast approach is sufficiently accurate, provided thatthe cosmologies are close to base ΛCDM (Rubino-Martın et al.2010; Shaw & Chluba 2011). Comparing the latest version ofrecfast (camb version) with CosmoRec, we find agreement towithin 0.2 % for hydrogen recombination (600 <∼ z <∼ 1600) and0.2 % during helium recombination for the standard ionizationhistory. The effect on the CMB power spectra is <∼ 0.15 % at ` <4000, although with slightly more pronounced shifts in the peakpositions than when comparing CosmoRec and HyRec. For thebase ΛCDM model, we find that the largest bias is on ns, at thelevel of 0.15σ (≈ 0.0006) for Planck TT,TE,EE+lowP+BAO.Although this is about 5 times larger than the difference in nsbetween CosmoRec and HyRec, this bias is nevertheless unim-portant at the current level of precision (and smaller than thedifferences seen from different likelihoods, see Sect. 3.1).

Finally we compare CosmoRec with recfast in its originalform (i.e., before recalibrating the fitting-functions on refinedrecombination calculations). For base ΛCDM, we expect to seebiases of ∆Ωbh2 ≈ −2.1σ ≈ −0.00028 and ∆ns ≈ −3.3σ ≈−0.012 (Shaw & Chluba 2011). Using the actual data (PlanckTT,TE,EE+lowP+BAO) we find biases of ∆Ωbh2 ≈ −1.8σ ≈−0.00024 and ∆ns ≈ −2.6σ ≈ −0.010, very close to the ex-pected values. This illustrates explicitly the importance of theimprovements of CosmoRec and HyRec over the original ver-sion of recfast for the interpretation of Planck data. However,CosmoRec and HyRec themselves are much more computation-ally intensive than the modified recfast, which is why we userecfast in most Planck cosmological analyses.

6.7.2. Measuring A2s→1s with Planck

The crucial role of the 2s–1s two-photon decay channel for thedynamics of hydrogen recombination has been appreciated sincethe early days of CMB research (Zeldovich et al. 1968; Peebles1968). Recombination is an out-of-equilibrium process and ener-getic photons emitted in the far Wien tail of the CMB by Lymancontinuum and series transitions keep the primordial plasmaionized for a much longer period than expected from simpleequilibrium recombination physics. Direct recombinations to theground state of hydrogen are prohibited, causing a modificationof the free electron number density, Ne, by only ∆Ne/Ne ≈ 10−6

around z ≈ 103 (Chluba & Sunyaev 2007). Similarly, the slowescape of photons from the Lyα resonance reduces the effec-tive Ly-α transition rate to A∗2p→1s ≈ 1–10 s−1 (by more thanseven orders of magnitude), making it comparable to the vacuum2s–1s two-photon decay rate of A2s→1s ≈ 8.22 s−1. About 57 %of all hydrogen atoms in the Universe became neutral throughthe 2s–1s channel (e.g., Wong et al. 2006; Chluba & Sunyaev

37Helium recombination is treated in more detail by CosmoRec (e.g.,Rubino-Martın et al. 2008; Chluba et al. 2012), which explains most ofthe difference.

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2006a), and subtle effects, such as the induced 2s–1s two-photondecay and Lyα re-absorption, need to be considered in pre-cision recombination calculations (Chluba & Sunyaev 2006b;Kholupenko & Ivanchik 2006; Hirata 2008).

The high sensitivity of the recombination process to the2s–1s two-photon transition rate also implies that instead ofsimply adopting a value for A2s→1s from theoretical computa-tions (Breit & Teller 1940; Spitzer & Greenstein 1951; Goldman1989) one can directly determine it with CMB data. From thetheoretical point of view it would be surprising to find a valuethat deviates significantly from A2s→1s = 8.2206 s−1, derivedfrom the most detailed computation (Labzowsky et al. 2005).However, laboratory measurements of this transition rate areextremely challenging (O’Connell et al. 1975; Kruger & Oed1975; Cesar et al. 1996). The most stringent limit is for the dif-ferential decay rate, A2s→1s(λ) dλ = (1.5±0.65) s−1 (a 43 % error)at wavelengths λ = 255.4–232.0 nm, consistent with the theoret-ical value of A2s→1s(λ) dλ = 1.02 s−1 in the same wavelengthrange (Kruger & Oed 1975). With precision data from Planckwe are in a position to perform the best measurement to date, us-ing cosmological data to inform us about atomic transition ratesat last scattering (as also emphasized by Mukhanov et al. 2012).

The 2s–1s two-photon rate affects the CMB anisotropiesonly through its effect on the recombination history. A largervalue of A2s→1s, accelerates recombination, allowing photonsand baryons to decouple earlier, an effect that shifts the acousticpeaks towards smaller scales. In addition, slightly less damp-ing occurs, as in the case of the stimulated 2s–1s two-photondecays (Chluba & Sunyaev 2006b). This implies that for flatcosmologies, variations of A2s→1s correlate with Ωch2 and H0(which affect the distance to the last scattering surface), whileA2s→1s anti-correlates with Ωbh2 and ns (which modify the slopeof the damping tail). Despite these degeneracies, one expectsthat Planck will provide a measurement of A2s→1s to within±0.5 s−1, corresponding to an approximately 6 % uncertainty(Mukhanov et al. 2012).

In Fig. 42, we show the marginalized posterior for A2s→1sfrom Planck and for Planck combined with BAO. UsingCosmoRec to compute the recombination history, we find

A2s→1s = 7.70 ± 1.01 s−1 Planck TT+lowP, (82a)A2s→1s = 7.72 ± 0.60 s−1 Planck TT,TE,EE+lowP, (82b)A2s→1s = 7.71 ± 0.99 s−1 Planck TT+lowP+BAO, (82c)A2s→1s = 7.75 ± 0.61 s−1 Planck TT,TE,EE+lowP

+BAO. (82d)

These results are in very good agreement with the theoreticalvalue, A2s→1s = 8.2206 s−1. For Planck TT,TE,EE+lowP+BAO,approximately 8 % precision is reached using cosmological data.These constraints are not sensitive to the addition of BAO, orother external data (JLA+H0). The slight shift away from thetheoretical value is accompanied by small (fractions of a σ)shifts in ns, Ωch2, and H0, to compensate for the effects of A2s→1son the distance to the last scattering surface and damping tail.This indicates that additional constraints on the acoustic scaleare required to fully break degeneracies between these param-eters and their effects on the CMB power spectrum, a task thatcould be achieved in the future using large-scale structure sur-veys and next generation CMB experiments.

The values for A2s→1s quoted above were obtained usingCosmoRec. When varying A2s→1s, the range of cosmologies be-comes large enough to potentially introduce a mismatch of therecfast fitting-functions that could affect the posterior. In par-ticular, with recfast the 2s–1s two-photon and Lyα channels

4.5 6.0 7.5 9.0 10.5

A2s→1s

0.0

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ma

x

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eory

CosmoRec TT+lowP+BAO

CosmoRec TT,TE,EE+lowP+BAO

RecFast TT,TE,EE+lowP+BAO

Fig. 42. Marginalized posterior for A2s→1s, obtained usingCosmoRec, with and without small-scale polarization data. Wefind good agreement with the theoretical value of A2s→1s =8.2206 s−1. For comparison, we also show the result for PlanckTT,TE,EE+lowP+BAO obtained with recfast, emphasizingthe consistency of different treatments.

are not treated separately, so that changes specific to the 2s–1s decay channel propagate inconsistently.38 However, repeatingthe analysis with recfast, we find A2s→1s = 7.78±0.58 s−1 (seeFig. 42), for Planck TT,TE,EE+lowP+BAO, which is in excel-lent agreement with CosmoRec, showing that these effects canbe neglected.

6.7.3. Measuring T0 at last-scattering with Planck

Our best constraint on the CMB monopole temperaturecomes from the measurements of the CMB spectrumwith COBE/FIRAS, giving a 0.02 % determination of T0(Fixsen et al. 1996; Fixsen 2009). Other constraints from molec-ular lines typically reach 1 % precision (see table 2 in Fixsen2009, for an overview), while independent BBN constraints pro-vide 5–10 % limits (Simha & Steigman 2008; Jeong et al. 2014).

The CMB anisotropies provide additional ways of deter-mining the value of T0 (for fixed values of Neff and YP). Oneis through the energy distribution of the CMB anisotropies(Fixsen et al. 1996; Fixsen 2003; Chluba 2014) and anotherthrough their power spectra (Opher & Pelinson 2004, 2005;Chluba 2014). Even small changes in T0, compatible with theCOBE/FIRAS error, affect the ionization history at the 0.5 %level around last-scattering, propagating to a roughly 0.1 %uncertainty in the CMB power spectrum (Chluba & Sunyaev2008). Overall, the effect of this uncertainty on the parametersof ΛCDM models is small (Hamann & Wong 2008); however,

38One effect is that by increasing A2s→1s fewer Lyα photons are pro-duced. This reduces the Lyα feedback correction to the 2s–1s channel,which further accelerates recombination, an effect that is not capturedwith recfast in the current implementation.

54


2.64 2.68 2.72 2.76 2.80

T0

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P/P

ma

x

FIR

AS

CosmoRec TT+lowP+BAO

CosmoRec TT,TE,EE+lowP+BAO

RecFast TT,TE,EE+lowP+BAO

Fig. 43. Marginalized posterior for T0. We find excellent agree-ment with the COBE/FIRAS measurement. For comparison, weshow the result for Planck TT,TE,EE+lowP+BAO obtained us-ing both CosmoRec and recfast, emphasizing the consistencyof different treatments.

without prior knowledge of T0 from the COBE/FIRAS measure-ment, the situation would change significantly.

The CMB monopole affects the CMB anisotropies in sev-eral ways. Most importantly, for larger T0, photons decouplefrom baryons at lower redshift, since more ionizing photons arepresent in the Wien-tail of the CMB. This effect is amplified be-cause of the exponential dependence of the atomic level popula-tions on the ratio of the ionization potentials and CMB tempera-ture. In addition, increasing T0 lowers the expansion timescale ofthe Universe and the redshift of matter-radiation equality, whileincreasing the photon sound speed. Some of these effects arealso produced by varying Neff ; however, the effects of T0 on theionization history and photon sound speed are distinct.

With CMB data alone, the determination of T0 is degeneratewith other parameters, but the addition of other data sets breaksthis degeneracy. Marginalized posterior distributions for T0 areshown in Fig. 43. Using CosmoRec, we find

T0 = 2.722 ± 0.027 K Planck TT+lowP+BAO, (83a)T0 = 2.718 ± 0.021 K Planck TT,TE,EE+lowP+BAO, (83b)

and similar results are obtained with recfast. This is in ex-cellent agreement with the COBE/FIRAS measurement, T0 =2.7255±0.0006 K (Fixsen et al. 1996; Fixsen 2009). These mea-surements of T0 reach a precision that is comparable to the ac-curacy obtained with interstellar molecules. Since the systemat-ics of these independent methods are very different, this resultdemonstrates the consistency of all these data. Allowing T0 tovary causes the errors of the other cosmological parameters toincrease. The strongest effect is on θMC, which is highly degen-erate with T0. The error on θMC increases by a factor of roughly25 if T0 is allowed to vary. The error on Ωbh2 increases by a fac-tor of about 4, while the errors on ns and Ωch2 increase by fac-tors of 1.5–2, and the other cosmological parameters are largely

unaffected by variations in T0. Because of the strong degener-acy with θMC, no constraint on T0 can be obtained using Planckdata alone. External data, such as BAO, are therefore required tobreak this geometric degeneracy.

It is important to emphasize that the CMB measures the tem-perature at a redshift of z ≈ 1100, so the comparison with mea-surements of T0 at the present day is effectively a test of theconstancy of aTCMB, where a ≈ 1/1100 is the scale-factor at thetime of last-scattering. It is remarkable that we are able to testthe constancy of aTCMB ≡ T0 over such a large dynamic rangein redshift. Of course, if we did find that aTCMB around recom-bination were discrepant with T0 now, then we would need toinvent a finely-tuned late-time photon injection mechanism39 toexplain the anomaly. Fortunately, the data are consistent with thestandard TCMB ∝ (1 + z) scaling of the CMB temperature.

Another approach to measuring aTCMB is through the ther-mal Sunyaev-Zeldovich effect in rich clusters of galaxies at var-ious redshifts (Fabbri et al. 1978; Rephaeli 1980), although it isunclear how one would interpret a failure of this test withoutan explicit model. In practice this approach is consistent witha scaling aTCMB = constant, but with lower precision than ob-tained here from Planck (e.g., Battistelli et al. 2002; Luzzi et al.2009; Saro et al. 2014; Hurier et al. 2014). A simple TCMB =T0(1 + z)1−β modification to the standard temperature redshiftrelation is frequently discussed in the literature (though this caseis not justified by any physical model and is difficult to realizewithout creating a CMB spectral distortion, see Chluba 2014).For this parameterization we find

β = (0.2 ± 1.4) × 10−3 Planck TT+lowP+BAO, (84a)β = (0.4 ± 1.1) × 10−3 Planck TT,TE,EE+lowP+BAO, (84b)

where we have adopted a recombination redshift of z∗ = 1100.40

Because of the long lever-arm in redshift afforded by the CMB,this is an improvement over earlier constraints by more than anorder of magnitude (e.g., Hurier et al. 2014).

In a self-consistent picture, changes of T0 would also affectthe BBN era. We might therefore consider a simultaneous varia-tion of Neff and YP to reflect the variation of the neutrino energydensity accompanying a putative variation in the photon energydensity. Since we find aTCMB at recombination to be highly con-sistent with the observed CMB temperature from COBE/FIRAS,considering this extra variation seems unnecessary. Instead, wemay view the aTCMB variation investigated here as further sup-port for the limits discussed in Sects. 6.4 and 6.5.

6.7.4. Semi-blind perturbed recombination analysis

The high sensitivity of small-scale CMB anisotropies to theionization history of the Universe around the epoch of recom-bination allows us to constrain possible deviations from thestandard recombination scenario in a model-independent way(Farhang et al. 2012, 2013). The method relies on an eigen-analysis (often referred to as a principle component analysis)of perturbations in the free electron fraction, Xe(z) = Ne/NH,where NH denotes the number density of hydrogen nuclei. Theeigenmodes selected are specific to the data used in the analysis.Similar approaches have been used to constrain deviations of thereionization history from the simplest models (Mortonson & Hu

39Pure energy release in the form of heating of ordinary matter wouldleave a Compton y-distortion (Zeldovich & Sunyaev 1969) at these latetimes (Burigana et al. 1991; Hu & Silk 1993; Chluba & Sunyaev 2012).

40The test depends on the logarithm of the redshift and so is insensi-tive to the precise value adopted for z∗.

55


Table 7. Standard parameters and the first three Xe-modes, asdetermined for Planck TT,TE,EE+lowP+BAO.

Parameter + 1 mode + 2 modes + 3 modes

Ωbh2 . . . 0.02229 ± 0.00017 0.02237 ± 0.00018 0.02237 ± 0.00019Ωch2 . . . 0.1190 ± 0.0010 0.1186 ± 0.0011 0.1187 ± 0.0012H0 . . . . 67.64 ± 0.48 67.80 ± 0.51 67.80 ± 0.56τ . . . . . . 0.065 ± 0.012 0.068 ± 0.013 0.068 ± 0.013ns . . . . . 0.9667 ± 0.0053 0.9677 ± 0.0055 0.9678 ± 0.0067ln(1010As) 3.062 ± 0.023 3.066 ± 0.024 3.066 ± 0.024µ1 . . . . . −0.03 ± 0.12 0.03 ± 0.14 0.02 ± 0.15µ2 . . . . . . . . −0.17 ± 0.18 −0.18 ± 0.19µ3 . . . . . . . . . . . −0.02 ± 0.88

2008) and annihilating dark matter scenarios (Finkbeiner et al.2012), both with the prior assumption that the standard recombi-nation physics is fully understood, as well as for constraining tra-jectories in inflation Planck Collaboration XX (2016) and darkenergy Planck Collaboration XIV (2016) parameterizations.

Here, we use Planck data to find preferred ionization frac-tion trajectories Xe(z) composed of low-order perturbation eigen-modes to the standard history (Xe-modes). The Xe-modes areconstructed through the eigen-decomposition of the inverse ofthe Fisher information matrix for base ΛCDM (the six cosmo-logical parameters and the nuisance parameters) and recombi-nation perturbation parameters (see Farhang et al. 2012, for de-tails). This procedure allows us to estimate the errors on theeigenmode amplitudes, µi, providing a rank ordering of the Xe-modes and their information content.

The first three Xe-modes for Planck TT,TE,EE+lowP are il-lustrated in Fig. 44, together with their impact on the differ-ential visibility function. Figure 45 shows the response of theCMB temperature and polarization power spectra to these eigen-modes. The first mode mainly leads to a change in the widthand height of the Thomson visibility function (bottom panel ofFig. 44). This implies less diffusion damping, which is also re-flected in the modifications to the CMB power spectra (as shownin Fig. 45). The second mode causes the visibility maximum toshift towards higher redshifts for µ2 > 0, which leads to a shiftof the CMB extrema to smaller scales; however, for roughlyconstant width of the visibility function it also introduces lessdamping at small scales. The third mode causes a combinationof changes in both the position and width of the visibility func-tion, with a pronounced effect on the location of the acousticpeaks. For the analysis of Planck data combinations, we onlyuse Xe-modes that are optimized for Planck TT,TE,EE+lowP.

We modified CosmoMC to estimate the mode amplitudes.The results for Planck TT,TE,EE+lowP+BAO are presented inTable 7. Although all mode amplitudes are consistent with stan-dard recombination, adding the second Xe-mode causes mildshifts in H0 and τ. For Planck TT+lowP, we find µ1 = −0.11 ±0.51 and µ2 = −0.23 ± 0.50, using the Planck TT,TE,EE+lowPeigenmodes, again consistent with the standard recombinationscenario. Adding the polarization data improves the errors bymore than a factor of 2. However, the mode amplitudes are in-sensitive to the addition of external data.

With pre-Planck data, only the amplitude, µ1, of the firsteigenmode could be constrained. The corresponding changein the ionization history translates mainly into a change inthe slope of the CMB damping tail, with this mode resem-bling the first mode determined using Planck data (Fig. 44).The WMAP9+SPT data gave a non-zero value for the first

500 1000 1500 2000 2500z

−0.05

0.00

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0.10

∆ln

Xe

Mode 1

Mode 2

Mode 3

800 1000 1200 1400z

−0.02

−0.01

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0.02

∆(v

isib

ility

)

Mode 1

Mode 2

Mode 3

Fig. 44. Eigen-modes of the recombination history, marginalizedover the standard six cosmological and Planck nuisance parame-ters. The upper panel shows the first three Xe-modes constructedfor Planck TT,TE,EE+lowP data. The lower panel show changesin the differential visibility corresponding to 1σ deviations fromthe standard recombination scenario for the first three Xe-modes.The maximum of the Thomson visibility function and width areindicated in both figures.

eigenmode at about 2σ, µSPT1 = −0.80 ± 0.37. However, the

WMAP9+ACT data gave µACT1 = 0.14 ± 0.45 and the com-

bined pre-Planck data (WMAP+ACT+SPT) gave µpre1 = −0.44±

0.33, both consistent with the standard recombination scenario(Calabrese et al. 2013). The variation among these results is an-other manifestation of the tensions between different pre-PlanckCMB data, as discussed in PCP13.

Although not optimal for Planck data, we also com-pute the amplitudes of the first three Xe-modes constructedfor the WMAP9+SPT data set. This provides a more di-rect comparison with the pre-Planck constraints. For PlanckTT,TE,EE+lowP+BAO we obtain µSPT

1 = −0.10 ± 0.13 andµSPT

2 = −0.13 ± 0.18. The mild tension of the pre-Planck datawith the standard recombination scenario disappears when us-ing Planck data. This is especially impressive, since the er-

56


1000 2000 3000`

−0.02

−0.01

0.00

0.01

0.02

∆ln

CT

T`

Mode 1

Mode 2

Mode 3

1000 2000 3000`

−0.02

−0.01

0.00

0.01

0.02

∆ln

CE

E`

Mode 1

Mode 2

Mode 3

Fig. 45. Changes in the TT (upper panel) and EE (lower panel)power spectra caused by a 1σ deviation from the standard re-combination scenario for the first three Xe-modes (see Fig. 44).

rors have improved by more than a factor of 2. By projectingonto the Planck modes, we find that the first two SPT modescan be expressed as µSPT

1 ≈ 0.69µ1 + 0.66µ2 ≈ −0.09 andµSPT

2 ≈ −0.70µ1 + 0.64µ2 ≈ −0.13, which emphasizes theconsistency of the results. Adding the first three SPT modes,we obtain µSPT

1 = −0.09 ± 0.13, µSPT2 = −0.14 ± 0.21, and

µSPT3 = −0.12 ± 0.86, which again is consistent with the stan-

dard model of recombination. The small changes in the modeamplitudes when adding the third mode arise because the SPTmodes are not optimal for Planck and so are correlated.

6.8. Cosmic defects

Topological defects are a generic by-product of symmetry-breaking phase transitions and a common phenomenon in con-densed matter systems. Cosmic defects of various types canbe formed in phase transitions in the early Universe (Kibble1976). In particular, cosmic strings can be produced in some su-persymmetric and grand-unified theories at the end of inflation(Jeannerot et al. 2003), as well as in higher-dimensional theories

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08f10

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max

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AH

SL

TX

Fig. 46. Marginalized posterior distributions for the fractionalcontribution, f10, of the defect contribution to the temperaturepower spectrum at ` = 10 (see the text for the precise defini-tion). Here we show the constraints for the Nambu-Goto cosmicstrings (NG, solid black), field-theory simulations of Abelian-Higgs cosmic strings (AH, solid red), semi-local strings (SL,dotted blue), and global textures (TX, dashed green). The upperpanel shows the 1D posterior for the Planck+lowP data, whileconstraints shown in the lower panel additionally use the T Eand EE data.

(e.g., Polchinski 2005). Constraints on the abundance of cos-mic strings and other defects therefore place limits on a rangeof models of the early Universe. More discussion on the forma-tion, evolution, and cosmological role of topological defects canbe found, for example, in the reviews by Vilenkin & Shellard(2000), Hindmarsh & Kibble (1995), and Copeland & Kibble(2010).

In this section we revisit the power spectrum-based con-straints on the abundance of cosmic strings and other topo-logical defects using the 2015 Planck data, including Planckpolarization measurements. The general approach follows thatdescribed in the Planck 2013 analysis of cosmic defects(Planck Collaboration XXV 2014), so here we focus on the up-dated constraints rather than on details of the methodology.

57


Topological defects are non-perturbative excitations of theunderlying field theory and their study requires numerical simu-lations. Unfortunately, since the Hubble scale, c/H0, is over 50orders of magnitude greater that the thickness of a GUT-scalestring, approximately (~/µc)1/2 with µ the mass per unit lengthof the string, it is impractical to simulate the string dynamics ex-actly in the late Universe. For this reason one needs to make ap-proximations. One approach considers the limit of an infinitelythin string, which corresponds to using the Nambu-Goto (“NG”)action for the string dynamics. In an alternative approach, theactual field dynamics for a given model are solved on a lattice.In this case it is necessary to resolve the string core, which gen-erally requires more computationally intensive simulations thanin the NG approach. Lattice simulations, however, can includeadditional physics, such as field radiation that is not present inNG simulations. Here we will use field-theory simulations ofthe Abelian-Higgs action (“AH”); details of these simulationsare discussed in Bevis et al. (2007, 2010).

The field-theory approach also allows one to simulate theo-ries in which the defects are not cosmic strings and so cannot bedescribed by the NG action. Examples include semi-local strings(“SL”, Urrestilla et al. 2008) and global defects. Here we willspecifically consider the breaking of a global O(4) symmetry re-sulting in texture defects (“TX”).

For the field-theory defects, we measure the energy-momentum tensor from the simulations and insert it as an ad-ditional constituent into a modified version of the CMBEASYBoltzmann code (Doran 2005) to predict the defect contribu-tion to the CMB temperature and polarization power spec-tra (see, e.g., Durrer et al. 2002). The same approach can beapplied to NG strings, but rather than using simulations di-rectly, we model the strings using the unconnected segmentmodel (“USM”, Albrecht et al. 1999; Pogosian & Vachaspati1999). In this model, strings are represented by a set of un-correlated straight segments, with scaling properties chosen tomatch those determined from numerical simulations. In thiscase, the string energy-momentum tensor can be computed ana-lytically and used as an active source in a modified Boltzmanncode. For this analysis we use CMBACT version 4,41 whereasPlanck Collaboration XXV (2014) used version 3. There havebeen several improvements to the code since the 2013 analysis,including a correction to the normalization of vector mode spec-tra. However, the largest change comes from an improved treat-ment of the scaling properties. The string correlation length andvelocity are described by an updated velocity-dependent one-scale model (Martins & Shellard 2002), which provides betteragreement with numerical simulations. Small-scale structure ofthe string, which was previously a free parameter, is accountedfor by the one-scale model.

The CMB power spectra from defects are proportional to(Gµ/c2)2. We scale the computed template CMB spectra, andadd these to the inflationary and foreground power spectra, toform the theory spectra that enter the likelihood. In practice, weparameterize the defects with their relative contribution to theTT spectrum at multipole ` = 10, f10 ≡ CTT (defect)

10 /CTT (total)10 .

We vary f10 and the standard six parameters of the base ΛCDMmodel, using CosmoMC. We also report our results in terms of thederived parameter Gµ/c2.

The constraints on f10 and the inferred limits on Gµ/c2 aresummarized in Table 8. The marginalized 1D posterior distribu-tion functions are shown in Fig. 46. For Planck TT+lowP wefind that the results are similar to the Planck+WP constraints re-

41http://www.sfu.ca/˜levon/cmbact.html

Table 8. 95 % upper limits on the parameter f10 and on the de-rived parameter Gµ/c2 for the defect models discussed in thetext. We show results for Planck TT+lowP data as well as forPlanck TT,TE,EE+lowP.

TT+lowP TT,TE,EE+lowPDefect type f10 Gµ/c2 f10 Gµ/c2

NG . . . . . . . < 0.020 < 1.8 × 10−7 < 0.011 < 1.3 × 10−7

AH . . . . . . . < 0.030 < 3.3 × 10−7 < 0.015 < 2.4 × 10−7

SL . . . . . . . < 0.039 < 10.6 × 10−7 < 0.024 < 8.5 × 10−7

TX . . . . . . . < 0.047 < 9.8 × 10−7 < 0.036 < 8.6 × 10−7

ported in Planck Collaboration XXV (2014), for the AH model,or somewhat better for SL and TX. However, the addition of thePlanck high-` T E and EE polarization data leads to a significantimprovement compared to the 2013 constraints.

For the NG string model, the results based on PlanckTT+lowP are slightly weaker than the 2013 Planck+WP con-straints. This is caused by a difference in the updated defectspectrum from the USM model, which has a less pronouncedpeak and shifts towards the AH spectrum. With the inclusion ofpolarization, Planck TT,TE,EE+lowP improves the upper limiton f10 by a factor of 2, as for the AH model. The differencesbetween the AH and NG results quoted here can be regardedas a rough indication of the uncertainty in the theoretical stringpower spectra.

In summary, we find no evidence for cosmic defects from thePlanck 2015 data, with tighter limits than before.

7. Conclusions42

(1) The six-parameter base ΛCDM model continues to providea very good match to the more extensive 2015 Planck data, in-cluding polarization. This is the most important conclusion ofthis paper.

(2) The 2015 Planck TT , T E, EE, and lensing spectra areconsistent with each other under the assumption of the baseΛCDM cosmology. However, when comparing the T E andEE spectra computed for different frequency combinations, wefind evidence for systematic effects caused by temperature-to-polarization leakage. These effects are at low levels and havelittle impact on the science conclusions of this paper.

(3) We have presented the first results on polarization fromthe LFI at low multipoles. The LFI polarization data, togetherwith Planck lensing and high-multipole temperature data, gives areionization optical depth of τ = 0.066±0.016 and a reionizationredshift of zre = 8.8+1.7

−1.4. These numbers are in good agreementwith those inferred from the WMAP9 polarization data cleanedfor polarized dust emission using HFI 353-GHz maps. They arealso in good agreement with results from Planck temperatureand lensing data, i.e., excluding any information from polariza-tion at low multipoles.

(4) The absolute calibration of the Planck 2015 HFI spectra ishigher by 2 % (in power) compared to 2013, largely resolving

42As in the abstract, here we quote 68 % confidence limits onmeasured parameters and 95 % upper limits on other parameters.

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the calibration difference noted in PCP13 between WMAP andPlanck. In addition, there have been a number of small changesto the low-level Planck processing and more accurate calibra-tions of the HFI beams. The 2015 Planck likelihood also makesmore aggressive use of sky than in PCP13 and incorporatessome refinements to the modelling of unresolved foregrounds.Apart from differences in τ (caused by switching to the LFI low-multipole polarization likelihood, as described in item 3 above)and the amplitude-τ combination Ase−2τ (caused by the changein absolute calibration), the 2015 parameters for base ΛCDM arein good agreement with those reported in PCP13.

(5) The Planck TT , T E, and EE spectra are accurately de-scribed by a purely adiabatic spectrum of fluctuations with aspectral tilt ns = 0.968 ± 0.006, consistent with the predic-tions of single-field inflationary models. Combining Planck datawith BAO, we find tight limits on the spatial curvature of theUniverse, |ΩK | < 0.005, again consistent with the inflationaryprediction of a spatially-flat Universe.

(6) The Planck data show no evidence for tensor modes. Addinga tensor amplitude as a one-parameter extension to base ΛCDM,we derive a 95 % upper limit of r0.002 < 0.11. This is consis-tent with the B-mode polarization analysis reported in BKP, re-solving the apparent discrepancy between the Planck constraintson r and the BICEP2 results reported by BICEP2 Collaboration(2014). In fact, by combining the Planck and BKP likelihoods,we find an even tighter constraint, r0.002 < 0.09, strongly dis-favouring inflationary models with a V(φ) ∝ φ2 potential.

(7) The Planck data show no evidence for any significant run-ning of the spectral index. We also set strong limits on a possibledeparture from a purely adiabatic spectrum, either through an ad-mixture of fully-correlated isocurvature modes or from cosmicdefects.

(8) The Planck best-fit base ΛCDM cosmology (we quote num-bers for Planck TT+lowP+lensing here) is in good agreementwith results from BAO surveys, and with the recent JLA sam-ple of Type Ia SNe. The Hubble constant in this cosmologyis H0 = (67.8 ± 0.9) km s−1Mpc−1, consistent with the di-rect measurement of H0 of Eq. (30) used as an H0 prior inthis paper. The Planck base ΛCDM cosmology is also con-sistent with the recent analysis of redshift-space distortions ofthe BOSS CMASS-DR11 data by Samushia et al. (2014) andBeutler et al. (2014). The amplitude of the present-day fluc-tuation spectrum, σ8, of the Planck base ΛCDM cosmologyis higher than inferred from weak lensing measurements fromthe CFHTLenS survey (Heymans et al. 2012; Erben et al. 2013)and, possibly, from counts of rich clusters of galaxies (includingPlanck cluster counts reported in Planck Collaboration XXIV2016). The Planck base ΛCDM cosmology is also discordantwith Lyα BAO measurements at z ≈ 2.35 (Delubac et al. 2015;Font-Ribera et al. 2014). At present, the reasons for these ten-sions are unclear.

(9) By combining the Planck TT+lowP+lensing data with otherastrophysical data, including the JLA supernovae, the equationof state for dark energy is constrained to w = −1.006 ± 0.045and is therefore compatible with a cosmological constant, as as-sumed in the base ΛCDM cosmology.

(10) We have presented a detailed analysis of possible ex-tensions to the neutrino sector of the base ΛCDM model.Combining Planck TT+lowP+lensing with BAO we find Neff =3.15 ± 0.23 for the effective number of relativistic degrees offreedom, consistent with the value Neff = 3.046 of the standardmodel. The sum of neutrino masses is constrained to

∑mν <

0.23 eV. The Planck data strongly disfavour fully thermalizedsterile neutrinos with msterile ≈ 1 eV that have been proposed as asolution to reactor neutrino oscillation anomalies. From Planck,we find no evidence for new neutrino physics. Standard neutri-nos with masses larger than those in the minimal mass hierarchyare still allowed, and could be detectable in combination withfuture astrophysical and CMB lensing data.

(11) The standard theory of big bang nucleosynthesis, withNeff = 3.046 and negligible leptonic asymmetry in the elec-tron neutrino sector, is in excellent agreement with Planck dataand observations of primordial light element abundances. Thisagreement is particularly striking for deuterium, for which accu-rate primordial abundance measurements have been reported re-cently (Cooke et al. 2014). The BBN theoretical predictions fordeuterium are now dominated by uncertainties in nuclear reac-tion rates (principally the d(p, γ)3He radiative capture process),rather than from Planck uncertainties in the physical baryon den-sity ωb ≡ Ωbh2.

(12) We have investigated the temperature and polarization sig-natures associated with annihilating dark matter and possible de-viations from the standard recombination history. Again, we findno evidence for new physics from the Planck data.

In summary, the Planck temperature and polarization spec-tra presented in Figs. 1 and 3 are more precise (and accu-rate) than those from any previous CMB experiment, and im-prove on the 2013 spectra presented in PCP13. Yet we find nosigns for any significant deviation from the base ΛCDM cos-mology. Similarly, the analysis of 2015 Planck data reportedin Planck Collaboration XVII (2016) sets unprecedentedly tightlimits on primordial non-Gaussianity. The Planck results of-fer powerful evidence in favour of simple inflationary mod-els, which provide an attractive mechanism for generating theslightly tilted spectrum of (nearly) Gaussian adiabatic perturba-tions that match our data to such high precision. In addition, thePlanck data show that the neutrino sector of the theory is con-sistent with the assumptions of the base ΛCDM model and thatthe dark energy is compatible with a cosmological constant. Ifthere is new physics beyond base ΛCDM, then the correspond-ing observational signatures in the CMB are weak and difficultto detect. This is the legacy of the Planck mission for cosmology.

Acknowledgements. The Planck Collaboration acknowledges the support of:ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF(Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, JA,and RES (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG (Germany);CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN(Norway); SFI (Ireland); FCT/MCTES (Portugal); ERC and PRACE (EU). Adescription of the Planck Collaboration and a list of its members, indicatingwhich technical or scientific activities they have been involved in, can be foundat http://www.cosmos.esa.int/web/planck/planck-collaboration.Some of the results in this paper have been derived using the HEALPix package.The research leading to these results has received funding from the EuropeanResearch Council under the European Union’s Seventh Framework Programme(FP/2007–2013) / ERC Grant Agreement No. [616170] and from the UKScience and Technology Facilities Council [grant number ST/L000652/1].

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Part of this work was undertaken on the STFC DiRAC HPC Facilities at theUniversity of Cambridge, funded by UK BIS National E-infrastructure capitalgrants, and on the Andromeda cluster of the University of Geneva. A large setof cosmological parameter constraints from different data combinations, andincluding many separate extensions to the 6-parameter base ΛCDM model, areavailable at http://pla.esac.esa.int/pla/#cosmology.

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1 APC, AstroParticule et Cosmologie, Universite Paris Diderot,CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne ParisCite, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex13, France

2 Aalto University Metsahovi Radio Observatory and Dept of RadioScience and Engineering, P.O. Box 13000, FI-00076 AALTO,Finland

3 African Institute for Mathematical Sciences, 6-8 Melrose Road,Muizenberg, Cape Town, South Africa

4 Agenzia Spaziale Italiana Science Data Center, Via del Politecnicosnc, 00133, Roma, Italy

5 Aix Marseille Universite, CNRS, LAM (Laboratoired’Astrophysique de Marseille) UMR 7326, 13388, Marseille,France

6 Aix Marseille Universite, Centre de Physique Theorique, 163Avenue de Luminy, 13288, Marseille, France

7 Astrophysics Group, Cavendish Laboratory, University ofCambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.K.

8 Astrophysics & Cosmology Research Unit, School of Mathematics,Statistics & Computer Science, University of KwaZulu-Natal,Westville Campus, Private Bag X54001, Durban 4000, SouthAfrica

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10 CGEE, SCS Qd 9, Lote C, Torre C, 4 andar, Ed. Parque CidadeCorporate, CEP 70308-200, Brasılia, DF, Brazil

11 CITA, University of Toronto, 60 St. George St., Toronto, ON M5S3H8, Canada

12 CNRS, IRAP, 9 Av. colonel Roche, BP 44346, F-31028 Toulousecedex 4, France

13 CRANN, Trinity College, Dublin, Ireland14 California Institute of Technology, Pasadena, California, U.S.A.15 Centre for Theoretical Cosmology, DAMTP, University of

Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K.16 Centro de Estudios de Fısica del Cosmos de Aragon (CEFCA),

Plaza San Juan, 1, planta 2, E-44001, Teruel, Spain17 Computational Cosmology Center, Lawrence Berkeley National

Laboratory, Berkeley, California, U.S.A.18 Consejo Superior de Investigaciones Cientıficas (CSIC), Madrid,

Spain19 DSM/Irfu/SPP, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex,

France20 DTU Space, National Space Institute, Technical University of

Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark21 Departement de Physique Theorique, Universite de Geneve, 24,

Quai E. Ansermet,1211 Geneve 4, Switzerland22 Departamento de Astrofısica, Universidad de La Laguna (ULL),

E-38206 La Laguna, Tenerife, Spain23 Departamento de Fısica, Universidad de Oviedo, Avda. Calvo

Sotelo s/n, Oviedo, Spain

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38 Dipartimento di Fisica e Astronomia G. Galilei, Universita degliStudi di Padova, via Marzolo 8, 35131 Padova, Italy

39 Dipartimento di Fisica e Scienze della Terra, Universita di Ferrara,Via Saragat 1, 44122 Ferrara, Italy

40 Dipartimento di Fisica, Universita La Sapienza, P. le A. Moro 2,Roma, Italy

41 Dipartimento di Fisica, Universita degli Studi di Milano, ViaCeloria, 16, Milano, Italy

42 Dipartimento di Fisica, Universita degli Studi di Trieste, via A.Valerio 2, Trieste, Italy

43 Dipartimento di Matematica, Universita di Roma Tor Vergata, Viadella Ricerca Scientifica, 1, Roma, Italy

44 Discovery Center, Niels Bohr Institute, Blegdamsvej 17,Copenhagen, Denmark

45 Discovery Center, Niels Bohr Institute, Copenhagen University,Blegdamsvej 17, Copenhagen, Denmark

46 European Southern Observatory, ESO Vitacura, Alonso de Cordova3107, Vitacura, Casilla 19001, Santiago, Chile

47 European Space Agency, ESAC, Planck Science Office, Caminobajo del Castillo, s/n, Urbanizacion Villafranca del Castillo,Villanueva de la Canada, Madrid, Spain

48 European Space Agency, ESTEC, Keplerlaan 1, 2201 AZNoordwijk, The Netherlands

49 Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100L’Aquila, Italy

50 HGSFP and University of Heidelberg, Theoretical PhysicsDepartment, Philosophenweg 16, 69120, Heidelberg, Germany

51 Haverford College Astronomy Department, 370 Lancaster Avenue,Haverford, Pennsylvania, U.S.A.

52 Helsinki Institute of Physics, Gustaf Hallstromin katu 2, Universityof Helsinki, Helsinki, Finland

53 INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78,Catania, Italy

54 INAF - Osservatorio Astronomico di Padova, Vicolodell’Osservatorio 5, Padova, Italy

55 INAF - Osservatorio Astronomico di Roma, via di Frascati 33,Monte Porzio Catone, Italy

66







56 INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11,Trieste, Italy

57 INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy58 INAF/IASF Milano, Via E. Bassini 15, Milano, Italy59 INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna,

Italy60 INFN, Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy61 INFN, Sezione di Roma 1, Universita di Roma Sapienza, Piazzale

Aldo Moro 2, 00185, Roma, Italy62 INFN, Sezione di Roma 2, Universita di Roma Tor Vergata, Via

della Ricerca Scientifica, 1, Roma, Italy63 INFN/National Institute for Nuclear Physics, Via Valerio 2,

I-34127 Trieste, Italy64 IPAG: Institut de Planetologie et d’Astrophysique de Grenoble,

Universite Grenoble Alpes, IPAG, F-38000 Grenoble, France,CNRS, IPAG, F-38000 Grenoble, France

65 ISDC, Department of Astronomy, University of Geneva, ch.d’Ecogia 16, 1290 Versoix, Switzerland

66 IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune411 007, India

67 Imperial College London, Astrophysics group, BlackettLaboratory, Prince Consort Road, London, SW7 2AZ, U.K.

68 Infrared Processing and Analysis Center, California Institute ofTechnology, Pasadena, CA 91125, U.S.A.

69 Institut Neel, CNRS, Universite Joseph Fourier Grenoble I, 25 ruedes Martyrs, Grenoble, France

70 Institut Universitaire de France, 103, bd Saint-Michel, 75005,Paris, France

71 Institut d’Astrophysique Spatiale, CNRS, Univ. Paris-Sud,Universite Paris-Saclay, Bat. 121, 91405 Orsay cedex, France

72 Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bisBoulevard Arago, F-75014, Paris, France

73 Institut fur Theoretische Teilchenphysik und Kosmologie, RWTHAachen University, D-52056 Aachen, Germany

74 Institute for Space Sciences, Bucharest-Magurale, Romania75 Institute of Astronomy, University of Cambridge, Madingley Road,

Cambridge CB3 0HA, U.K.76 Institute of Theoretical Astrophysics, University of Oslo, Blindern,

Oslo, Norway77 Instituto de Astrofısica de Canarias, C/Vıa Lactea s/n, La Laguna,

Tenerife, Spain78 Instituto de Fısica de Cantabria (CSIC-Universidad de Cantabria),

Avda. de los Castros s/n, Santander, Spain79 Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via

Marzolo 8, I-35131 Padova, Italy80 Jet Propulsion Laboratory, California Institute of Technology, 4800

Oak Grove Drive, Pasadena, California, U.S.A.81 Jodrell Bank Centre for Astrophysics, Alan Turing Building,

School of Physics and Astronomy, The University of Manchester,Oxford Road, Manchester, M13 9PL, U.K.

82 Kavli Institute for Cosmological Physics, University of Chicago,Chicago, IL 60637, USA

83 Kavli Institute for Cosmology Cambridge, Madingley Road,Cambridge, CB3 0HA, U.K.

84 Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008,Russia

85 LAL, Universite Paris-Sud, CNRS/IN2P3, Orsay, France86 LERMA, CNRS, Observatoire de Paris, 61 Avenue de

l’Observatoire, Paris, France87 Laboratoire AIM, IRFU/Service d’Astrophysique - CEA/DSM -

CNRS - Universite Paris Diderot, Bat. 709, CEA-Saclay, F-91191Gif-sur-Yvette Cedex, France

88 Laboratoire Traitement et Communication de l’Information, CNRS(UMR 5141) and Telecom ParisTech, 46 rue Barrault F-75634Paris Cedex 13, France

89 Laboratoire de Physique Subatomique et Cosmologie, UniversiteGrenoble-Alpes, CNRS/IN2P3, 53, rue des Martyrs, 38026Grenoble Cedex, France

90 Laboratoire de Physique Theorique, Universite Paris-Sud 11 &CNRS, Batiment 210, 91405 Orsay, France

91 Lawrence Berkeley National Laboratory, Berkeley, California,U.S.A.

92 Lebedev Physical Institute of the Russian Academy of Sciences,Astro Space Centre, 84/32 Profsoyuznaya st., Moscow, GSP-7,117997, Russia

93 Leung Center for Cosmology and Particle Astrophysics, NationalTaiwan University, Taipei 10617, Taiwan

94 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1,85741 Garching, Germany

95 McGill Physics, Ernest Rutherford Physics Building, McGillUniversity, 3600 rue University, Montreal, QC, H3A 2T8, Canada

96 National University of Ireland, Department of ExperimentalPhysics, Maynooth, Co. Kildare, Ireland

97 Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716Warsaw, Poland

98 Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark99 Niels Bohr Institute, Copenhagen University, Blegdamsvej 17,

Copenhagen, Denmark100 Nordita (Nordic Institute for Theoretical Physics),

Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden101 Optical Science Laboratory, University College London, Gower

Street, London, U.K.102 Physics Department, Shahid Beheshti University, Tehran, Iran103 SISSA, Astrophysics Sector, via Bonomea 265, 34136, Trieste,

Italy104 SMARTEST Research Centre, Universita degli Studi e-Campus,

Via Isimbardi 10, Novedrate (CO), 22060, Italy105 School of Physics and Astronomy, Cardiff University, Queens

Buildings, The Parade, Cardiff, CF24 3AA, U.K.106 School of Physics and Astronomy, University of Nottingham,

Nottingham NG7 2RD, U.K.107 Sorbonne Universite-UPMC, UMR7095, Institut d’Astrophysique

de Paris, 98 bis Boulevard Arago, F-75014, Paris, France108 Space Research Institute (IKI), Russian Academy of Sciences,

Profsoyuznaya Str, 84/32, Moscow, 117997, Russia109 Space Sciences Laboratory, University of California, Berkeley,

California, U.S.A.110 Special Astrophysical Observatory, Russian Academy of Sciences,

Nizhnij Arkhyz, Zelenchukskiy region, Karachai-CherkessianRepublic, 369167, Russia

111 Stanford University, Dept of Physics, Varian Physics Bldg, 382 ViaPueblo Mall, Stanford, California, U.S.A.

112 Sub-Department of Astrophysics, University of Oxford, KebleRoad, Oxford OX1 3RH, U.K.

113 Sydney Institute for Astronomy, School of Physics A28, Universityof Sydney, NSW 2006, Australia

114 The Oskar Klein Centre for Cosmoparticle Physics, Department ofPhysics,Stockholm University, AlbaNova, SE-106 91 Stockholm,Sweden

115 Theory Division, PH-TH, CERN, CH-1211, Geneva 23,Switzerland

116 UPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago,F-75014, Paris, France

117 Universite de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex4, France

118 Universities Space Research Association, StratosphericObservatory for Infrared Astronomy, MS 232-11, Moffett Field,CA 94035, U.S.A.

119 University Observatory, Ludwig Maximilian University of Munich,Scheinerstrasse 1, 81679 Munich, Germany

120 University of Granada, Departamento de Fısica Teorica y delCosmos, Facultad de Ciencias, Granada, Spain

121 University of Granada, Instituto Carlos I de Fısica Teorica yComputacional, Granada, Spain

122 University of Heidelberg, Institute for Theoretical Physics,Philosophenweg 16, 69120, Heidelberg, Germany

123 Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478Warszawa, Poland

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Planck 2015 results. XIII. Cosmological parameters · 2016-06-20 · Astronomy & Astrophysics manuscript no. planck˙parameters˙2015 c ESO 2016 June 20, 2016 Planck 2015 results